(* Author: Johannes Hölzl <hoelzl@in.tum.de> *)

section ‹Auxiliary Theory›

text ‹Parts of it should be moved to the Isabelle repository›

theory Markov_Models_Auxiliary
imports
  "HOL-Probability.Probability"
  "HOL-Library.Rewrite"
  "HOL-Library.Linear_Temporal_Logic_on_Streams"
  Coinductive.Coinductive_Stream
  Coinductive.Coinductive_Nat
begin

lemma lfp_upperbound: "(⋀y. x ≤ f y) ⟹ x ≤ lfp f"
  unfolding lfp_def by (intro Inf_greatest) (auto intro: order_trans)

(* ?? *)
lemma lfp_arg: "(λt. lfp (F t)) = lfp (λx t. F t (x t))"
  apply (auto simp: lfp_def le_fun_def fun_eq_iff intro!: Inf_eqI Inf_greatest)
  subgoal for x y
    by (rule INF_lower2[of "top(x := y)"]) auto
  done

lemma lfp_pair: "lfp (λf (a, b). F (λa b. f (a, b)) a b) (a, b) = lfp F a b"
  unfolding lfp_def
  by (auto intro!: INF_eq simp: le_fun_def)
     (auto intro!: exI[of _ "λ(a, b). x a b" for x])

lemma all_Suc_split: "(∀i. P i) ⟷ (P 0 ∧ (∀i. P (Suc i)))"
  using nat_induct by auto

definition "with P f d = (if ∃x. P x then f (SOME x. P x) else d)"

lemma withI[case_names default exists]:
  "((⋀x. ¬ P x) ⟹ Q d) ⟹ (⋀x. P x ⟹ Q (f x)) ⟹ Q (with P f d)"
  unfolding with_def by (auto intro: someI2)

context order
begin

definition
  "maximal f S = {x∈S. ∀y∈S. f y ≤ f x}"

lemma maximalI: "x ∈ S ⟹ (⋀y. y ∈ S ⟹ f y ≤ f x) ⟹ x ∈ maximal f S"
  by (simp add: maximal_def)

lemma maximalI_trans: "x ∈ maximal f S ⟹ f x ≤ f y ⟹ y ∈ S ⟹ y ∈ maximal f S"
  unfolding maximal_def by (blast intro: antisym order_trans)

lemma maximalD1: "x ∈ maximal f S ⟹ x ∈ S"
  by (simp add: maximal_def)

lemma maximalD2: "x ∈ maximal f S ⟹ y ∈ S ⟹ f y ≤ f x"
  by (simp add: maximal_def)

lemma maximal_inject: "x ∈ maximal f S ⟹ y ∈ maximal f S ⟹ f x = f y"
  unfolding maximal_def by (blast intro: antisym)

lemma maximal_empty[simp]: "maximal f {} = {}"
  by (simp add: maximal_def)

lemma maximal_singleton[simp]: "maximal f {x} = {x}"
  by (auto simp add: maximal_def)

lemma maximal_in_S: "maximal f S ⊆ S"
  by (auto simp: maximal_def)

end

context linorder
begin

lemma maximal_ne:
  assumes "finite S" "S ≠ {}"
  shows "maximal f S ≠ {}"
  using assms
proof (induct rule: finite_ne_induct)
  case (insert s S)
  show ?case
  proof cases
    assume "∀x∈S. f x ≤ f s"
    with insert have "s ∈ maximal f (insert s S)"
      by (auto intro!: maximalI)
    then show ?thesis
      by auto
  next
    assume "¬ (∀x∈S. f x ≤ f s)"
    then have "maximal f (insert s S) = maximal f S"
      by (auto simp: maximal_def)
    with insert show ?thesis
      by auto
  qed
qed simp

end

lemma mono_les:
  fixes s S N and l1 l2 :: "'a ⇒ real" and K :: "'a ⇒ 'a pmf"
  defines "Δ x ≡ l2 x - l1 x"
  assumes s: "s ∈ S" and S: "(⋃s∈S. set_pmf (K s)) ⊆ S ∪ N"
  assumes int_l1[simp]: "⋀s. s ∈ S ⟹ integrable (K s) l1"
  assumes int_l2[simp]: "⋀s. s ∈ S ⟹ integrable (K s) l2"
  assumes to_N: "⋀s. s ∈ S ⟹ ∃t∈N. (s, t) ∈ (SIGMA s:UNIV. K s)⇧*"
  assumes l1: "⋀s. s ∈ S ⟹ (∫t. l1 t ∂K s) + c s ≤ l1 s"
  assumes l2: "⋀s. s ∈ S ⟹ l2 s ≤ (∫t. l2 t ∂K s) + c s"
  assumes eq: "⋀s. s ∈ N ⟹ l2 s ≤ l1 s"
  assumes finitary: "finite (Δ ` (S∪N))"
  shows "l2 s ≤ l1 s"
proof -
  define M where "M = {s∈S∪N. ∀t∈S∪N. Δ t ≤ Δ s}"

  have [simp]: "⋀s. s∈S ⟹ integrable (K s) Δ"
    by (simp add: Δ_def[abs_def])

  have M_unqiue: "⋀s t. s ∈ M ⟹ t ∈ M ⟹ Δ s = Δ t"
    by (auto intro!: antisym simp: M_def)
  have M1: "⋀s. s ∈ M ⟹ s ∈ S ∪ N"
    by (auto simp: M_def)
  have M2: "⋀s t. s ∈ M ⟹ t ∈ S ∪ N ⟹ Δ t ≤ Δ s"
    by (auto simp: M_def)
  have M3: "⋀s t. s ∈ M ⟹ t ∈ S ∪ N ⟹ t ∉ M ⟹ Δ t < Δ s"
    by (auto simp: M_def less_le)

  have N: "∀s∈N. Δ s ≤ 0"
    using eq by (simp add: Δ_def)

  { fix s assume s: "s ∈ M" "M ∩ N = {}"
    then have "s ∈ S - N"
      by (auto dest: M1)
    with to_N[of s] obtain t where "(s, t) ∈ (SIGMA s:UNIV. K s)⇧*" and "t ∈ N"
      by (auto simp: M_def)
    from this(1) ‹s ∈ M› have "Δ s ≤ 0"
    proof (induction rule: converse_rtrancl_induct)
      case (step s s')
      then have s: "s ∈ M" "s ∈ S" "s ∉ N" and s': "s' ∈ S ∪ N" "s' ∈ K s"
        using S ‹M ∩ N = {}› by (auto dest: M1)
      have "s' ∈ M"
      proof (rule ccontr)
        assume "s' ∉ M"
        with ‹s ∈ S› s' ‹s ∈ M›
        have "0 < pmf (K s) s'" "Δ s' < Δ s"
          by (auto intro: M2 M3 pmf_positive)

        have "Δ s ≤ ((∫t. l2 t ∂K s) + c s) - ((∫t. l1 t ∂K s) + c s)"
          unfolding Δ_def using ‹s ∈ S› ‹s ∉ N› by (intro diff_mono l1 l2) auto
        then have "Δ s ≤ (∫s'. Δ s' ∂K s)"
          using ‹s ∈ S› by (simp add: Δ_def)
        also have "… < (∫s'. Δ s ∂K s)"
          using ‹s' ∈ K s› ‹Δ s' < Δ s› ‹s∈S› S ‹s∈M›
          by (intro measure_pmf.integral_less_AE[where A="{s'}"])
             (auto simp: emeasure_measure_pmf_finite AE_measure_pmf_iff set_pmf_iff[symmetric]
                   intro!: M2)
        finally show False
          using measure_pmf.prob_space[of "K s"] by simp
      qed
      with step.IH ‹t∈N› N have "Δ s' ≤ 0" "s' ∈ M"
        by auto
      with ‹s∈S› show "Δ s ≤ 0"
        by (force simp: M_def)
    qed (insert N ‹t∈N›, auto) }

  show ?thesis
  proof cases
    assume "M ∩ N = {}"
    have "Max (Δ`(S∪N)) ∈ Δ`(S∪N)"
      using ‹s ∈ S› by (intro Max_in finitary) auto
    then obtain t where "t ∈ S ∪ N" "Δ t = Max (Δ`(S∪N))"
      unfolding image_iff by metis
    then have "t ∈ M"
      by (auto simp: M_def finitary intro!: Max_ge)
    have "Δ s ≤ Δ t"
      using ‹t∈M› ‹s∈S› by (auto dest: M2)
    also have "Δ t ≤ 0"
      using ‹t∈M› ‹M ∩ N = {}› by fact
    finally show ?thesis
      by (simp add: Δ_def)
  next
    assume "M ∩ N ≠ {}"
    then obtain t where "t ∈ M" "t ∈ N" by auto
    with N ‹s∈S› have "Δ s ≤ 0"
      by (intro order_trans[of "Δ s" "Δ t" 0]) (auto simp: M_def)
    then show ?thesis
      by (simp add: Δ_def)
  qed
qed

lemma unique_les:
  fixes s S N and l1 l2 :: "'a ⇒ real" and K :: "'a ⇒ 'a pmf"
  defines "Δ x ≡ l2 x - l1 x"
  assumes s: "s ∈ S" and S: "(⋃s∈S. set_pmf (K s)) ⊆ S ∪ N"
  assumes "⋀s. s ∈ S ⟹ integrable (K s) l1"
  assumes "⋀s. s ∈ S ⟹ integrable (K s) l2"
  assumes "⋀s. s ∈ S ⟹ ∃t∈N. (s, t) ∈ (SIGMA s:UNIV. K s)⇧*"
  assumes "⋀s. s ∈ S ⟹ l1 s = (∫t. l1 t ∂K s) + c s"
  assumes "⋀s. s ∈ S ⟹ l2 s = (∫t. l2 t ∂K s) + c s"
  assumes "⋀s. s ∈ N ⟹ l2 s = l1 s"
  assumes 1: "finite (Δ ` (S∪N))"
  shows "l2 s = l1 s"
proof -
  have "finite ((λx. l2 x - l1 x) ` (S∪N))"
    using 1 by (auto simp: Δ_def[abs_def])
  moreover then have "finite (uminus ` (λx. l2 x - l1 x) ` (S∪N))"
    by auto
  ultimately show ?thesis
    using assms
    by (intro antisym mono_les[of s S K N l2 l1 c] mono_les[of s S K N l1 l2 c])
       (auto simp: image_comp comp_def)
qed

lemma inf_continuous_suntil_disj[order_continuous_intros]:
  assumes Q: "inf_continuous Q"
  assumes disj: "⋀x ω. ¬ (P ω ∧ Q x ω)"
  shows "inf_continuous (λx. P suntil Q x)"
  unfolding inf_continuous_def
proof (safe intro!: ext)
  fix M ω i assume "(P suntil Q (⨅i. M i)) ω" "decseq M" then show "(P suntil Q (M i)) ω"
    unfolding inf_continuousD[OF Q ‹decseq M›] by induction (auto intro: suntil.intros)
next
  fix M ω assume *: "(⨅i. P suntil Q (M i)) ω" "decseq M"
  then have "(P suntil Q (M 0)) ω"
    by auto
  from this * show "(P suntil Q (⨅i. M i)) ω"
    unfolding inf_continuousD[OF Q ‹decseq M›]
  proof induction
    case (base ω) with disj[of ω "M _"] show ?case by (auto intro: suntil.intros elim: suntil.cases)
  next
    case (step ω) with disj[of ω "M _"] show ?case by (auto intro: suntil.intros elim: suntil.cases)
  qed
qed

lemma inf_continuous_nxt[order_continuous_intros]: "inf_continuous P ⟹ inf_continuous (λx. nxt (P x) ω)"
  by (auto simp: inf_continuous_def image_comp)

lemma sup_continuous_nxt[order_continuous_intros]: "sup_continuous P ⟹ sup_continuous (λx. nxt (P x) ω)"
  by (auto simp: sup_continuous_def image_comp)

lemma mcont_ennreal_of_enat: "mcont Sup (≤) Sup (≤) ennreal_of_enat"
  by (auto intro!: mcontI monotoneI contI ennreal_of_enat_Sup)

lemma mcont2mcont_ennreal_of_enat[cont_intro]:
  "mcont lub ord Sup (≤) f ⟹ mcont lub ord Sup (≤) (λx. ennreal_of_enat (f x))"
  by (auto intro: ccpo.mcont2mcont[OF complete_lattice_ccpo'] mcont_ennreal_of_enat)

declare stream.exhaust[cases type: stream]

lemma scount_eq_emeasure: "scount P ω = emeasure (count_space UNIV) {i. P (sdrop i ω)}"
proof cases
  assume "alw (ev P) ω"
  moreover then have "infinite {i. P (sdrop i ω)}"
    using infinite_iff_alw_ev[of P ω] by simp
  ultimately show ?thesis
    by (simp add: scount_infinite_iff[symmetric])
next
  assume "¬ alw (ev P) ω"
  moreover then have "finite {i. P (sdrop i ω)}"
    using infinite_iff_alw_ev[of P ω] by simp
  ultimately show ?thesis
    by (simp add: not_alw_iff not_ev_iff scount_eq_card)
qed

lemma measurable_scount[measurable]:
  assumes [measurable]: "Measurable.pred (stream_space M) P"
  shows "scount P ∈ measurable (stream_space M) (count_space UNIV)"
  unfolding scount_eq[abs_def] by measurable

lemma measurable_sfirst2:
  assumes [measurable]: "Measurable.pred (N ⨂⇩M stream_space M) (λ(x, ω). P x ω)"
  shows "(λ(x, ω). sfirst (P x) ω) ∈ measurable (N ⨂⇩M  stream_space M) (count_space UNIV)"
  apply (coinduction rule: measurable_enat_coinduct)
  apply simp
  apply (rule exI[of _ "λx. 0"])
  apply (rule exI[of _ "λ(x, ω). (x, stl ω)"])
  apply (rule exI[of _ "λ(x, ω). P x ω"])
  apply (subst sfirst.simps[abs_def])
  apply (simp add: fun_eq_iff)
  done

lemma measurable_sfirst2'[measurable (raw)]:
  assumes [measurable (raw)]: "f ∈ N →⇩M stream_space M" "Measurable.pred (N ⨂⇩M stream_space M) (λx. P (fst x) (snd x))"
  shows "(λx. sfirst (P x) (f x)) ∈ measurable N (count_space UNIV)"
  using measurable_sfirst2[measurable] by measurable

lemma measurable_sfirst[measurable]:
  assumes [measurable]: "Measurable.pred (stream_space M) P"
  shows "sfirst P ∈ measurable (stream_space M) (count_space UNIV)"
  by measurable

lemma measurable_epred[measurable]: "epred ∈ count_space UNIV →⇩M count_space UNIV"
  by (rule measurable_count_space)

lemma nn_integral_stretch:
  "f ∈ borel →⇩M borel ⟹ c ≠ 0 ⟹ (∫⇧+x. f (c * x) ∂lborel) = (1 / ¦c¦::real) * (∫⇧+x. f x ∂lborel)"
  using nn_integral_real_affine[of f c 0] by (simp add: mult.assoc[symmetric] ennreal_mult[symmetric])

lemma prod_sum_distrib:
  fixes f g :: "'a ⇒ 'b ⇒ 'c::comm_semiring_1"
  assumes "finite I" shows "(⋀i. i ∈ I ⟹ finite (J i)) ⟹ (∏i∈I. ∑j∈J i. f i j) = (∑m∈Pi⇩E I J. ∏i∈I. f i (m i))"
  using ‹finite I›
proof induction
  case (insert i I) then show ?case
    by (auto simp: PiE_insert_eq finite_PiE sum.reindex inj_combinator sum.swap[of _ "Pi⇩E I J"]
                   sum_cartesian_product' sum_distrib_left sum_distrib_right
             intro!: sum.cong prod.cong arg_cong[where f="(*) x" for x])
qed simp

lemma prod_add_distrib:
  fixes f g :: "'a ⇒ 'b::comm_semiring_1"
  assumes "finite I" shows "(∏i∈I. f i + g i) = (∑J∈Pow I. (∏i∈J. f i) * (∏i∈I - J. g i))"
proof -
  have "(∏i∈I. f i + g i) = (∏i∈I. ∑b∈{True, False}. if b then f i else g i)"
    by simp
  also have "… = (∑m∈I →⇩E {True, False}. ∏i∈I. if m i then f i else g i)"
    using ‹finite I› by (rule prod_sum_distrib) simp
  also have "… = (∑J∈Pow I. (∏i∈J. f i) * (∏i∈I - J. g i))"
    by (rule sum.reindex_bij_witness[where i="λJ. λi∈I. i∈J" and j="λm. {i∈I. m i}"])
       (auto simp: fun_eq_iff prod.If_cases ‹finite I› intro!: arg_cong2[where f="(*)"] prod.cong)
  finally show ?thesis .
qed

subclass (in linordered_nonzero_semiring) ordered_semiring_0
  proof qed

lemma (in linordered_nonzero_semiring) prod_nonneg: "(∀a∈A. 0 ≤ f a) ⟹ 0 ≤ prod f A"
  by (induct A rule: infinite_finite_induct) simp_all

lemma (in linordered_nonzero_semiring) prod_mono:
  "∀i∈A. 0 ≤ f i ∧ f i ≤ g i ⟹ prod f A ≤ prod g A"
  by (induct A rule: infinite_finite_induct) (auto intro!: prod_nonneg mult_mono)

lemma (in linordered_nonzero_semiring) prod_mono2:
  assumes "finite J" "I ⊆ J" "⋀i. i ∈ I ⟹ 0 ≤ g i ∧ g i ≤ f i" "(⋀i. i ∈ J - I ⟹ 1 ≤ f i)"
  shows "prod g I ≤ prod f J"
proof -
  have "prod g I = (∏i∈J. if i ∈ I then g i else 1)"
    using ‹finite J› ‹I ⊆ J› by (simp add: prod.If_cases Int_absorb1)
  also have "… ≤ prod f J"
    using assms by (intro prod_mono) auto
  finally show ?thesis .
qed

lemma (in linordered_nonzero_semiring) prod_mono3:
  assumes "finite J" "I ⊆ J" "⋀i. i ∈ J ⟹ 0 ≤ g i" "⋀i. i ∈ I ⟹ g i ≤ f i" "(⋀i. i ∈ J - I ⟹ g i ≤ 1)"
  shows "prod g J ≤ prod f I"
proof -
  have "prod g J ≤ (∏i∈J. if i ∈ I then f i else 1)"
    using assms by (intro prod_mono) auto
  also have "… = prod f I"
    using ‹finite J› ‹I ⊆ J› by (simp add: prod.If_cases Int_absorb1)
  finally show ?thesis .
qed

lemma (in linordered_nonzero_semiring) one_le_prod: "(⋀i. i ∈ I ⟹ 1 ≤ f i) ⟹ 1 ≤ prod f I"
proof (induction I rule: infinite_finite_induct)
  case (insert i I) then show ?case
    using mult_mono[of 1 "f i" 1 "prod f I"]
    by (auto intro: order_trans[OF zero_le_one])
qed auto

lemma sum_plus_one_le_prod_plus_one:
  fixes p :: "'a ⇒ 'b::linordered_nonzero_semiring"
  assumes "⋀i. i ∈ I ⟹ 0 ≤ p i"
  shows "(∑i∈I. p i) + 1 ≤ (∏i∈I. p i + 1)"
proof cases
  assume [simp]: "finite I"
  with assms have [simp]: "J ⊆ I ⟹ 0 ≤ prod p J" for J
    by (intro prod_nonneg) auto
  have "1 + (∑i∈I. p i) = (∑J∈insert {} ((λx. {x})`I). (∏i∈J. p i) * (∏i∈I - J. 1))"
    by (subst sum.insert) (auto simp: sum.reindex)
  also have "… ≤ (∑J∈Pow I. (∏i∈J. p i) * (∏i∈I - J. 1))"
    using assms by (intro sum_mono2) auto
  finally show ?thesis
    by (subst prod_add_distrib) (auto simp: add.commute)
qed simp

lemma summable_iff_convergent_prod:
  fixes p :: "nat ⇒ real" assumes p: "⋀i. 0 ≤ p i"
  shows "summable p ⟷ convergent (λn. ∏i<n. p i + 1)"
  unfolding summable_iff_convergent
proof
  assume "convergent (λn. ∏i<n. p i + 1)"
  then obtain x where x: "(λn. ∏i<n. p i + 1) ⇢ x"
    by (auto simp: convergent_def)
  then have "1 ≤ x"
    by (rule tendsto_lowerbound) (auto intro!: always_eventually one_le_prod p)

  have "convergent (λn. 1 + (∑i<n. p i))"
  proof (intro Bseq_mono_convergent BseqI allI)
    show "0 < x" using ‹1 ≤ x› by auto
  next
    fix n
    have "norm ((∑i<n. p i) + 1) ≤ (∏i<n. p i + 1)"
      using p by (simp add: sum_nonneg sum_plus_one_le_prod_plus_one p)
    also have "… ≤ x"
      using assms
      by (intro tendsto_lowerbound[OF x])
          (auto simp: eventually_sequentially intro!: exI[of _ n] prod_mono2)
    finally show "norm (1 + sum p {..<n}) ≤ x"
      by (simp add: add.commute)
  qed (insert p, auto intro!: sum_mono2)
  then show "convergent (λn. ∑i<n. p i)"
    unfolding convergent_add_const_iff .
next
  assume "convergent (λn. ∑i<n. p i)"
  then obtain x where x: "(λn. exp (∑i<n. p i)) ⇢ exp x"
    by (force simp: convergent_def intro!: tendsto_exp)
  show "convergent (λn. ∏i<n. p i + 1)"
  proof (intro Bseq_mono_convergent BseqI allI)
    show "0 < exp x" by simp
  next
    fix n
    have "norm (∏i<n. p i + 1) ≤ exp (∑i<n. p i)"
      using p exp_ge_add_one_self[of "p _"] by (auto simp add: prod_nonneg exp_sum add.commute intro!: prod_mono)
    also have "… ≤ exp x"
      using p
      by (intro tendsto_lowerbound[OF x]) (auto simp: eventually_sequentially intro!: sum_mono2 )
    finally show "norm (∏i<n. p i + 1) ≤ exp x" .
  qed (insert p, auto intro!: prod_mono2)
qed

primrec eexp :: "ereal ⇒ ennreal"
  where
    "eexp MInfty = 0"
  | "eexp (ereal r) = ennreal (exp r)"
  | "eexp PInfty = top"

lemma
  shows eexp_minus_infty[simp]: "eexp (-∞) = 0"
    and eexp_infty[simp]: "eexp ∞ = top"
  using eexp.simps by simp_all

lemma eexp_0[simp]: "eexp 0 = 1"
  by (simp add: zero_ereal_def)

lemma eexp_inj[simp]: "eexp x = eexp y ⟷ x = y"
  by (cases x; cases y; simp)

lemma eexp_mono[simp]: "eexp x ≤ eexp y ⟷ x ≤ y"
  by (cases x; cases y; simp add: top_unique)

lemma eexp_strict_mono[simp]: "eexp x < eexp y ⟷ x < y"
  by (simp add: less_le)

lemma exp_eq_0_iff[simp]: "eexp x = 0 ⟷ x = -∞"
  using eexp_inj[of x "-∞"] unfolding eexp_minus_infty .

lemma eexp_surj: "range eexp = UNIV"
proof -
  have part: "UNIV = {0} ∪ {0 <..< top} ∪ {top::ennreal}"
    by (auto simp: less_top)
  show ?thesis
    unfolding part
    by (force simp: image_iff less_top less_top_ennreal intro!: eexp.simps[symmetric] eexp.simps dest: exp_total)
qed

lemma continuous_on_eexp': "continuous_on UNIV eexp"
  by (rule continuous_onI_mono) (auto simp: eexp_surj)

lemma continuous_on_eexp[continuous_intros]: "continuous_on A f ⟹ continuous_on A (λx. eexp (f x))"
  by (rule continuous_on_compose2[OF continuous_on_eexp']) auto

lemma tendsto_eexp[tendsto_intros]: "(f ⤏ x) F ⟹ ((λx. eexp (f x)) ⤏ eexp x) F"
  by (rule continuous_on_tendsto_compose[OF continuous_on_eexp']) auto

lemma measurable_eexp[measurable]: "eexp ∈ borel →⇩M borel"
  using continuous_on_eexp' by (rule borel_measurable_continuous_onI)

lemma eexp_add: "¬ ((x = ∞ ∧ y = -∞) ∨ (x = -∞ ∧ y = ∞)) ⟹ eexp (x + y) = eexp x * eexp y"
  by (cases x; cases y; simp add: exp_add ennreal_mult ennreal_top_mult ennreal_mult_top)

lemma sum_Pinfty:
  fixes f :: "'a ⇒ ereal"
  shows "sum f I = ∞ ⟷ (finite I ∧ (∃i∈I. f i = ∞))"
  by (induction I rule: infinite_finite_induct) auto

lemma sum_Minfty:
  fixes f :: "'a ⇒ ereal"
  shows "sum f I = -∞ ⟷ (finite I ∧ ¬ (∃i∈I. f i = ∞) ∧ (∃i∈I. f i = -∞))"
  by (induction I rule: infinite_finite_induct)
     (auto simp: sum_Pinfty)

lemma eexp_sum: "¬ (∃i∈I. ∃j∈I. f i = -∞ ∧ f j = ∞) ⟹ eexp (∑i∈I. f i) = (∏i∈I. eexp (f i))"
proof (induction I rule: infinite_finite_induct)
  case (insert i I)
  have "eexp (sum f (insert i I)) = eexp (f i) * eexp (sum f I)"
    using insert.prems insert.hyps by (auto simp: sum_Pinfty sum_Minfty intro!: eexp_add)
  then show ?case
    using insert by auto
qed simp_all

lemma eexp_suminf:
  assumes wf_f: "¬ {-∞, ∞} ⊆ range f" and f: "summable f"
  shows "(λn. ∏i<n. eexp (f i)) ⇢ eexp (∑i. f i)"
proof -
  have "(λn. eexp (∑i<n. f i)) ⇢ eexp (∑i. f i)"
    by (intro tendsto_eexp summable_LIMSEQ f)
  also have "(λn. eexp (∑i<n. f i)) = (λn. ∏i<n. eexp (f i))"
    using wf_f by (auto simp: fun_eq_iff image_iff eq_commute intro!: eexp_sum)
  finally show ?thesis .
qed

lemma continuous_onI_antimono:
  fixes f :: "'a::linorder_topology ⇒ 'b::{dense_order,linorder_topology}"
  assumes "open (f`A)"
    and mono: "⋀x y. x ∈ A ⟹ y ∈ A ⟹ x ≤ y ⟹ f y ≤ f x"
  shows "continuous_on A f"
proof (rule continuous_on_generate_topology[OF open_generated_order], safe)
  have monoD: "⋀x y. x ∈ A ⟹ y ∈ A ⟹ f y < f x ⟹ x < y"
    by (auto simp: not_le[symmetric] mono)
  have "∃x. x ∈ A ∧ f x < b ∧ x < a" if a: "a ∈ A" and fa: "f a < b" for a b
  proof -
    obtain y where "f a < y" "{f a ..< y} ⊆ f`A"
      using open_right[OF ‹open (f`A)›, of "f a" b] a fa
      by auto
    obtain z where z: "f a < z" "z < min b y"
      using dense[of "f a" "min b y"] ‹f a < y› ‹f a < b› by auto
    then obtain c where "z = f c" "c ∈ A"
      using ‹{f a ..< y} ⊆ f`A›[THEN subsetD, of z] by (auto simp: less_imp_le)
    with a z show ?thesis
      by (auto intro!: exI[of _ c] simp: monoD)
  qed
  then show "∃C. open C ∧ C ∩ A = f -` {..<b} ∩ A" for b
    by (intro exI[of _ "(⋃x∈{x∈A. f x < b}. {x <..})"])
       (auto intro: le_less_trans[OF mono] less_imp_le)

  have "∃x. x ∈ A ∧ b < f x ∧ x > a" if a: "a ∈ A" and fa: "b < f a" for a b
  proof -
    note a fa
    moreover
    obtain y where "y < f a" "{y <.. f a} ⊆ f`A"
      using open_left[OF ‹open (f`A)›, of "f a" b]  a fa
      by auto
    then obtain z where z: "max b y < z" "z < f a"
      using dense[of "max b y" "f a"] ‹y < f a› ‹b < f a› by auto
    then obtain c where "z = f c" "c ∈ A"
      using ‹{y <.. f a} ⊆ f`A›[THEN subsetD, of z] by (auto simp: less_imp_le)
    with a z show ?thesis
      by (auto intro!: exI[of _ c] simp: monoD)
  qed
  then show "∃C. open C ∧ C ∩ A = f -` {b <..} ∩ A" for b
    by (intro exI[of _ "(⋃x∈{x∈A. b < f x}. {..< x})"])
       (auto intro: less_le_trans[OF _ mono] less_imp_le)
qed

lemma minus_add_eq_ereal: "¬ ((a = ∞ ∧ b = -∞) ∨ (a = -∞ ∧ b = ∞)) ⟹ - (a + b::ereal) = -a - b"
  by (cases a; cases b; simp)

lemma setsum_negf_ereal: "¬ {-∞, ∞} ⊆ f`I ⟹ (∑i∈I. - f i) = - (∑i∈I. f i::ereal)"
  by (induction I rule: infinite_finite_induct)
     (auto simp: minus_add_eq_ereal sum_Minfty sum_Pinfty,
      (subst minus_add_eq_ereal; auto simp: sum_Pinfty sum_Minfty image_iff minus_ereal_def)+)

lemma convergent_minus_iff_ereal: "convergent (λx. - f x::ereal) ⟷ convergent f"
  unfolding convergent_def  by (metis ereal_uminus_uminus ereal_Lim_uminus)

lemma summable_minus_ereal: "¬ {-∞, ∞} ⊆ range f ⟹ summable (λn. f n) ⟹ summable (λn. - f n::ereal)"
  unfolding summable_iff_convergent
  by (subst setsum_negf_ereal) (auto simp: convergent_minus_iff_ereal)

lemma (in product_prob_space) product_nn_integral_component:
  assumes "f ∈ borel_measurable (M i)""i ∈ I"
  shows "integral⇧N (Pi⇩M I M) (λx. f (x i)) = integral⇧N (M i) f"
proof -
  from assms show ?thesis
    apply (subst PiM_component[symmetric, OF ‹i ∈ I›])
    apply (subst nn_integral_distr[OF measurable_component_singleton])
    apply simp_all
    done
qed

lemma ennreal_inverse_le[simp]: "inverse x ≤ inverse y ⟷ y ≤ (x::ennreal)"
  by (cases "0 < x"; cases x; cases "0 < y"; cases y; auto simp: top_unique inverse_ennreal)

lemma inverse_inverse_ennreal[simp]: "inverse (inverse x::ennreal) = x"
  by (cases "0 < x"; cases x; auto simp: inverse_ennreal)

lemma range_inverse_ennreal: "range inverse = (UNIV::ennreal set)"
proof -
  have "∃x. y = inverse x" for y :: ennreal
    by (intro exI[of _ "inverse y"]) simp
  then show ?thesis
    unfolding surj_def by auto
qed

lemma continuous_on_inverse_ennreal': "continuous_on (UNIV :: ennreal set) inverse"
  by (rule continuous_onI_antimono) (auto simp: range_inverse_ennreal)

lemma sums_minus_ereal: "¬ {- ∞, ∞} ⊆ f ` UNIV ⟹ (λn. - f n::ereal) sums x ⟹ f sums - x"
  unfolding sums_def
  apply (subst ereal_Lim_uminus)
  apply (subst (asm) setsum_negf_ereal)
  apply auto
  done

lemma suminf_minus_ereal: "¬ {- ∞, ∞} ⊆ f ` UNIV ⟹ summable f ⟹ (∑n. - f n :: ereal) = - suminf f"
  apply (rule sums_unique[symmetric])
  apply (rule sums_minus_ereal)
  apply (auto simp: ereal_uminus_eq_reorder)
  done

end

(* Author: Johannes Hölzl <hoelzl@in.tum.de> *)

section ‹Discrete-Time Markov Chain›

theory Discrete_Time_Markov_Chain
  imports Markov_Models_Auxiliary
begin

text ‹

Markov chain with discrete time steps and discrete state space.

›

lemma sstart_eq': "sstart Ω (x # xs) = {ω. shd ω = x ∧ stl ω ∈ sstart Ω xs}"
  by (auto simp: sstart_eq)

lemma measure_eq_stream_space_coinduct[consumes 1, case_names left right cont]:
  assumes "R N M"
  assumes R_1: "⋀N M. R N M ⟹ N ∈ space (prob_algebra (stream_space (count_space UNIV)))"
    and R_2: "⋀N M. R N M ⟹ M ∈ space (prob_algebra (stream_space (count_space UNIV)))"
    and cont: "⋀N M. R N M ⟹ ∃N' M' p. (∀y∈set_pmf p. R (N' y) (M' y)) ∧
      (∀x. N' x ∈ space (prob_algebra (stream_space (count_space UNIV)))) ∧ (∀x. M' x ∈ space (prob_algebra (stream_space (count_space UNIV)))) ∧ 
      N = (measure_pmf p ⤜ (λy. distr (N' y) (stream_space (count_space UNIV)) ((##) y))) ∧
      M = (measure_pmf p ⤜ (λy. distr (M' y) (stream_space (count_space UNIV)) ((##) y)))"
  shows "N = M"
proof -
  let ?S = "stream_space (count_space UNIV)"
  have "∀N M. R N M ⟶ (∃N' M' p. (∀y∈set_pmf p. R (N' y) (M' y)) ∧
      (∀x. N' x ∈ space (prob_algebra ?S)) ∧ (∀x. M' x ∈ space (prob_algebra ?S)) ∧
      N = (measure_pmf p ⤜ (λy. distr (N' y) ?S ((##) y))) ∧
      M = (measure_pmf p ⤜ (λy. distr (M' y) ?S ((##) y))))"
    using cont by auto
  then obtain n m p where
    p: "⋀N M y. R N M ⟹ y ∈ set_pmf (p N M) ⟹ R (n N M y) (m N M y)" and
    n: "⋀N M x. R N M ⟹ n N M x ∈ space (prob_algebra ?S)" and
    n_eq: "⋀N M y. R N M ⟹ N = (measure_pmf (p N M) ⤜ (λy. distr (n N M y) ?S ((##) y)))" and
    m: "⋀N M x. R N M ⟹ m N M x ∈ space (prob_algebra ?S)" and
    m_eq: "⋀N M y. R N M ⟹ M = (measure_pmf (p N M) ⤜ (λy. distr (m N M y) ?S ((##) y)))"
    unfolding choice_iff' choice_iff by blast

  define A where "A = (SIGMA nm:UNIV. (λx. (n (fst nm) (snd nm) x, m (fst nm) (snd nm) x)) ` p (fst nm) (snd nm))"
  have A_singleton: "A `` {nm} = (λx. (n (fst nm) (snd nm) x, m (fst nm) (snd nm) x)) ` p (fst nm) (snd nm)" for nm
    by (auto simp: A_def)

  have sets_n[measurable_cong, simp]: "sets (n N M y) = sets ?S" if "R N M" for N M y
    using n[OF that, of y] by (auto simp: space_prob_algebra)
  have sets_m[measurable_cong, simp]: "sets (m N M y) = sets ?S" if "R N M" for N M y
    using m[OF that, of y] by (auto simp: space_prob_algebra)
  have [simp]: "R N M ⟹ prob_space (n N M y)" for N M y
    using n[of N M y] by (auto simp: space_prob_algebra)
  have [simp]: "R N M ⟹ prob_space (m N M y)" for N M y
    using m[of N M y] by (auto simp: space_prob_algebra)
  have [measurable]: "R N M ⟹ n N M ∈ count_space UNIV →⇩M subprob_algebra ?S" for N M
    by (rule measurable_prob_algebraD) (auto intro: n)
  have [measurable]: "R N M ⟹ m N M ∈ count_space UNIV →⇩M subprob_algebra ?S" for N M
    by (rule measurable_prob_algebraD) (auto intro: m)

  define n' where "n' N M y = distr (n N M y) ?S ((##) y)" for N M y
  define m' where "m' N M y = distr (m N M y) ?S ((##) y)" for N M y
  have n'_eq: "R N M ⟹ N = (measure_pmf (p N M) ⤜ n' N M)" for N M unfolding n'_def by (rule n_eq)
  have m'_eq: "R N M ⟹ M = (measure_pmf (p N M) ⤜ m' N M)" for N M unfolding m'_def by (rule m_eq)
  have [measurable]: "R N M ⟹ n' N M ∈ count_space UNIV →⇩M subprob_algebra ?S" for N M
    unfolding n'_def by (rule measurable_distr2[where M="?S"]) measurable
  have [measurable]: "R N M ⟹ m' N M ∈ count_space UNIV →⇩M subprob_algebra ?S" for N M
    unfolding m'_def by (rule measurable_distr2[where M="?S"]) measurable

  have n'_shd: "R N M ⟹ distr (n' N M y) (count_space UNIV) shd = measure_pmf (return_pmf y)" for N M y
    unfolding n'_def by (subst distr_distr) (auto simp: comp_def prob_space.distr_const return_pmf.rep_eq)
  have m'_shd: "R N M ⟹ distr (m' N M y) (count_space UNIV) shd = measure_pmf (return_pmf y)" for N M y
    unfolding m'_def by (subst distr_distr) (auto simp: comp_def prob_space.distr_const return_pmf.rep_eq)
  have n'_stl: "R N M ⟹ distr (n' N M y) ?S stl = n N M y" for N M y
    unfolding n'_def by (subst distr_distr) (auto simp: comp_def distr_id2)
  have m'_stl: "R N M ⟹ distr (m' N M y) ?S stl = m N M y" for N M y
    unfolding m'_def by (subst distr_distr) (auto simp: comp_def distr_id2)

  define F where "F = (A⇧* `` {(N, M)})"
  have "countable F"
    unfolding F_def
    apply (intro countable_rtrancl countable_insert[of _ "(N, M)"] countable_empty)
    apply (rule countable_Image)
     apply (auto simp: A_singleton)
    done
  have F_NM[simp]: "(N, M) ∈ F" unfolding F_def by auto
  have R_F[simp]: "R N' M'" if "(N', M') ∈ F" for N' M'
  proof -
    have "((N, M), (N', M')) ∈ A⇧*" using that by (auto simp: F_def)
    then show "R N' M'"
      by (induction p=="(N', M')" arbitrary: N' M' rule: rtrancl_induct) (auto simp: ‹R N M› A_def p)
  qed
  have nm_F: "(n N' M' y, m N' M' y) ∈ F" if "y ∈ p N' M'" "(N', M') ∈ F" for N' M' y
  proof -
    have *: "((N, M), (N', M')) ∈ A⇧*" using that by (auto simp: F_def)
    with that show ?thesis
      apply (simp add: F_def)
      apply (intro rtrancl.rtrancl_into_rtrancl[OF *])
      apply (auto simp: A_def)
      done
  qed

  define Ω where "Ω = (⋃(n, m)∈F. set_pmf (p n m))"
  have [measurable]: "Ω ∈ sets (count_space UNIV)" by auto
  have in_Ω: "(N, M) ∈ F ⟹ y ∈ p N M ⟹ y ∈ Ω" for N M y
    by (auto simp: Ω_def Bex_def)

  show ?thesis
  proof (intro stream_space_eq_sstart)
    from ‹countable F› show "countable Ω"
      by (auto simp add: Ω_def)
    show "prob_space N" "prob_space M" "sets N = sets ?S" "sets M = sets ?S"
      using R_1[OF ‹R N M›] R_2[OF ‹R N M›] by (auto simp add: space_prob_algebra)
    have "⋀N M. (N, M) ∈ F ⟹ AE x in N. x !! i ∈ Ω" for i
    proof (induction i)
      case 0 note NM = 0[THEN R_F, simp] show ?case
        apply (subst n'_eq[OF NM])
        apply (subst AE_bind[where B="?S"])
          apply measurable
        apply (auto intro!: AE_distrD[where f=shd and M'="count_space UNIV"]
                    simp: AE_measure_pmf_iff n[OF NM] n'_shd in_Ω[OF 0] cong: AE_cong_simp)
        done
    next
      case (Suc i) note NM = Suc(2)[THEN R_F, simp]
      show ?case
        apply (subst n'_eq[OF NM])
        apply (subst AE_bind[where B="?S"])
          apply measurable
        apply (auto intro!: AE_distrD[where f=stl and M'="?S"] Suc(1)[OF nm_F] Suc(2)
          simp: AE_measure_pmf_iff n'_stl cong: AE_cong_simp)
        done
    qed
    then have AE_N: "⋀N M. (N, M) ∈ F ⟹ AE x in N. x ∈ streams Ω"
      unfolding streams_iff_snth AE_all_countable by auto
    then show "AE x in N. x ∈ streams Ω" by (blast intro: F_NM)

    have "⋀N M. (N, M) ∈ F ⟹ AE x in M. x !! i ∈ Ω" for i
    proof (induction i arbitrary: N M)
      case 0 note NM = 0[THEN R_F, simp] show ?case
        apply (subst m'_eq[OF NM])
        apply (subst AE_bind[where B="?S"])
          apply measurable
        apply (auto intro!: AE_distrD[where f=shd and M'="count_space UNIV"]
                    simp: AE_measure_pmf_iff m[OF NM] m'_shd in_Ω[OF 0] cong: AE_cong_simp)
        done
    next
      case (Suc i) note NM = Suc(2)[THEN R_F, simp]
      show ?case
        apply (subst m'_eq[OF NM])
        apply (subst AE_bind[where B="?S"])
          apply measurable
        apply (auto intro!: AE_distrD[where f=stl and M'="?S"] Suc(1)[OF nm_F] Suc(2)
          simp: AE_measure_pmf_iff m'_stl cong: AE_cong_simp)
        done
    qed
    then have AE_M: "⋀N M. (N, M) ∈ F ⟹ AE x in M. x ∈ streams Ω"
      unfolding streams_iff_snth AE_all_countable by auto
    then show "AE x in M. x ∈ streams Ω" by (blast intro: F_NM)

    fix xs assume "xs ∈ lists Ω"
    with ‹(N, M) ∈ F› show "emeasure N (sstart Ω xs) = emeasure M (sstart Ω xs)"
    proof (induction xs arbitrary: N M)
      case Nil
      have "prob_space N" "prob_space M" "sets N = sets ?S" "sets M = sets ?S"
        using R_1[OF R_F[OF Nil(1)]] R_2[OF R_F[OF Nil(1)]] by (auto simp add: space_prob_algebra)
      have "emeasure N (streams Ω) = 1"
        by (rule prob_space.emeasure_eq_1_AE[OF ‹prob_space N› _ AE_N[OF Nil(1)]])
           (auto simp add: ‹sets N = sets ?S› intro!: streams_sets)
      moreover have "emeasure M (streams Ω) = 1"
        by (rule prob_space.emeasure_eq_1_AE[OF ‹prob_space M› _ AE_M[OF Nil(1)]])
           (auto simp add: ‹sets M = sets ?S› intro!: streams_sets)
      ultimately show ?case by simp
    next
      case (Cons x xs)
      note NM = Cons(2)[THEN R_F, simp]
      have *: "(##) y -` sstart Ω (x # xs) = (if x = y then sstart Ω xs else {})" for y
        by auto
      show ?case
        apply (subst n'_eq[OF NM])
        apply (subst (3) m'_eq[OF NM])
        apply (subst emeasure_bind[OF _ _ sstart_sets])
          apply simp []
         apply measurable []
        apply (subst emeasure_bind[OF _ _ sstart_sets])
          apply simp []
         apply measurable []
        apply (intro nn_integral_cong_AE AE_pmfI)
        apply (subst n'_def)
        apply (subst m'_def)
        using Cons(3)
        apply (auto intro!: Cons nm_F
          simp add: emeasure_distr sets_eq_imp_space_eq[OF sets_n] sets_eq_imp_space_eq[OF sets_m]
                    space_stream_space *)
        done
    qed
  qed
qed

subsection ‹Discrete Markov Kernel›

locale MC_syntax =
  fixes K :: "'s ⇒ 's pmf"
begin

abbreviation acc :: "('s × 's) set" where
  "acc ≡ (SIGMA s:UNIV. K s)⇧*"

abbreviation acc_on :: "'s set ⇒ ('s × 's) set" where
  "acc_on S ≡ (SIGMA s:UNIV. K s ∩ S)⇧*"

lemma countable_reachable: "countable (acc `` {s})"
  by (auto intro!: countable_rtrancl countable_set_pmf simp: Sigma_Image)

lemma countable_acc: "countable X ⟹ countable (acc `` X)"
  apply (rule countable_Image)
  apply (rule countable_reachable)
  apply assumption
  done

context
  notes [[inductive_internals]]
begin

coinductive enabled where
  "enabled (shd ω) (stl ω) ⟹ shd ω ∈ K s ⟹ enabled s ω"

end

lemma alw_enabled: "enabled (shd ω) (stl ω) ⟹ alw (λω. enabled (shd ω) (stl ω)) ω"
  by (coinduction arbitrary: ω rule: alw_coinduct) (auto elim: enabled.cases)

abbreviation "S ≡ stream_space (count_space UNIV)"

lemma in_S [measurable (raw)]: "x ∈ space S"
  by (simp add: space_stream_space)

inductive_simps enabled_iff: "enabled s ω"

lemma enabled_Stream: "enabled x (y ## ω) ⟷ y ∈ K x ∧ enabled y ω"
  by (subst enabled_iff)  auto

lemma measurable_enabled[measurable]:
  "Measurable.pred (stream_space (count_space UNIV)) (enabled s)" (is "Measurable.pred ?S _")
  unfolding enabled_def
proof (coinduction arbitrary: s rule: measurable_gfp2_coinduct)
  case (step A s)
  then have [measurable]: "⋀t. Measurable.pred ?S (A t)" by auto
  have *: "⋀x. (∃ω t. s = t ∧ x = ω ∧ A (shd ω) (stl ω) ∧ shd ω ∈ set_pmf (K t)) ⟷
    (∃t∈K s. A t (stl x) ∧ t = shd x)"
    by auto
  note countable_set_pmf[simp]
  show ?case
    unfolding * by measurable
qed (auto simp: inf_continuous_def)

lemma enabled_iff_snth: "enabled s ω ⟷ (∀i. ω !! i ∈ K ((s ## ω) !! i))"
proof safe
  fix i assume "enabled s ω" then show "ω !! i ∈ K ((s ## ω) !! i)"
    by (induct i arbitrary: s ω)
       (force elim: enabled.cases)+
next
  assume "∀i. ω !! i ∈ set_pmf (K ((s ## ω) !! i))" then show "enabled s ω"
    by (coinduction arbitrary: s ω)
       (auto elim: allE[of _ "Suc i" for i] allE[of _ 0])
qed

primcorec force_enabled where
  "force_enabled x ω =
    (let y = if shd ω ∈ K x then shd ω else (SOME y. y ∈ K x) in y ## force_enabled y (stl ω))"

lemma force_enabled_in_set_pmf[simp, intro]: "shd (force_enabled x ω) ∈ K x"
  by (auto simp: some_in_eq set_pmf_not_empty)

lemma enabled_force_enabled: "enabled x (force_enabled x ω)"
  by (coinduction arbitrary: x ω) (auto simp: some_in_eq set_pmf_not_empty)

lemma force_enabled: "enabled x ω ⟹ force_enabled x ω = ω"
  by (coinduction arbitrary: x ω) (auto elim: enabled.cases)

lemma Ex_enabled: "∃ω. enabled x ω"
  by (rule exI[of _ "force_enabled x undefined"] enabled_force_enabled)+

lemma measurable_force_enabled: "force_enabled x ∈ measurable S S"
proof (rule measurable_stream_space2)
  fix n show "(λω. force_enabled x ω !! n) ∈ measurable S (count_space UNIV)"
  proof (induction n arbitrary: x)
    case (Suc n) show ?case
      apply simp
      apply (rule measurable_compose_countable'[OF measurable_compose[OF measurable_stl Suc], where I="set_pmf (K x)"])
      apply (rule measurable_compose[OF measurable_shd])
      apply (auto simp: countable_set_pmf some_in_eq set_pmf_not_empty)
      done
  qed (auto intro!: measurable_compose[OF measurable_shd])
qed

abbreviation "D ≡ stream_space (Π⇩M s∈UNIV. K s)"

lemma sets_D: "sets D = sets (stream_space (Π⇩M s∈UNIV. count_space UNIV))"
  by (intro sets_stream_space_cong sets_PiM_cong) simp_all

lemma space_D: "space D = space (stream_space (Π⇩M s∈UNIV. count_space UNIV))"
  using sets_eq_imp_space_eq[OF sets_D] .

lemma measurable_D_D: "measurable D D =
    measurable (stream_space (Π⇩M s∈UNIV. count_space UNIV)) (stream_space (Π⇩M s∈UNIV. count_space UNIV))"
  by (simp add: measurable_def space_D sets_D)

primcorec walk :: "'s ⇒ ('s ⇒ 's) stream ⇒ 's stream" where
  "shd (walk s ω) = (if shd ω s ∈ K s then shd ω s else (SOME t. t ∈ K s))"
| "stl (walk s ω) = walk (if shd ω s ∈ K s then shd ω s else (SOME t. t ∈ K s)) (stl ω)"

lemma enabled_walk: "enabled s (walk s ω)"
  by (coinduction arbitrary: s ω) (auto simp: some_in_eq set_pmf_not_empty)

lemma measurable_walk[measurable]: "walk s ∈ measurable D S"
proof -
  note measurable_compose[OF measurable_snth, intro!]
  note measurable_compose[OF measurable_component_singleton, intro!]
  note if_weak_cong[cong del]
  note measurable_g = measurable_compose_countable'[OF _ _ countable_reachable]

  define n :: nat where "n = 0"
  define g where "g = (λ_::('s ⇒ 's) stream. s)"
  then have "g ∈ measurable D (count_space (acc `` {s}))"
    by auto
  then have "(λx. walk (g x) (sdrop n x)) ∈ measurable D S"
  proof (coinduction arbitrary: g n rule: measurable_stream_coinduct)
    case (shd g) show ?case
      by (fastforce intro: measurable_g[OF _ shd])
  next
    case (stl g) show ?case
      by (fastforce simp add: sdrop.simps[symmetric] some_in_eq set_pmf_not_empty
                    simp del: sdrop.simps intro: rtrancl_into_rtrancl measurable_g[OF _ stl])
  qed
  then show ?thesis
    by (simp add: g_def n_def)
qed

subsection ‹Trace Space for Discrete-Time Markov Chains›

definition T :: "'s ⇒ 's stream measure" where
  "T s = distr (stream_space (Π⇩M s∈UNIV. K s)) S (walk s)"

lemma space_T[simp]: "space (T s) = space S"
  by (simp add: T_def)

lemma sets_T[simp, measurable_cong]: "sets (T s) = sets S"
  by (simp add: T_def)

lemma measurable_T1[simp]: "measurable (T s) M = measurable S M"
  by (intro measurable_cong_sets) simp_all

lemma measurable_T2[simp]: "measurable M (T s) = measurable M S"
  by (intro measurable_cong_sets) simp_all

lemma in_measurable_T1[measurable (raw)]: "f ∈ measurable S M ⟹ f ∈ measurable (T s) M"
  by simp

lemma in_measurable_T2[measurable (raw)]: "f ∈ measurable M S ⟹ f ∈ measurable M (T s)"
  by simp

lemma AE_T_enabled: "AE ω in T s. enabled s ω"
  unfolding T_def by (simp add: AE_distr_iff enabled_walk)

sublocale T: prob_space "T s" for s
proof -
  interpret P: product_prob_space K UNIV ..
  interpret prob_space "stream_space (Π⇩M s∈UNIV. K s)"
    by (rule P.prob_space_stream_space)
  fix s show "prob_space (T s)"
    by (simp add: T_def prob_space_distr)
qed

lemma emeasure_T_const[simp]: "emeasure (T s) (space S) = 1"
  using T.emeasure_space_1[of s] by simp

lemma nn_integral_T:
  assumes f[measurable]: "f ∈ borel_measurable S"
  shows "(∫⇧+X. f X ∂T s) = (∫⇧+t. (∫⇧+ω. f (t ## ω) ∂T t) ∂K s)"
proof -
  interpret product_prob_space K UNIV ..
  interpret D: prob_space "stream_space (Π⇩M s∈UNIV. K s)"
    by (rule prob_space_stream_space)

  have T: "⋀f s. f ∈ borel_measurable S ⟹ (∫⇧+X. f X ∂T s) = (∫⇧+ω. f (walk s ω) ∂D)"
    by (simp add: T_def nn_integral_distr)

  have "(∫⇧+X. f X ∂T s) = (∫⇧+ω. f (walk s ω) ∂D)"
    by (rule T) measurable
  also have "… = (∫⇧+d. ∫⇧+ω. f (walk s (d ## ω)) ∂D ∂Π⇩M i∈UNIV. K i)"
    by (simp add: P.nn_integral_stream_space)
  also have "… = (∫⇧+d. (∫⇧+ω. f (d s ## walk (d s) ω) * indicator {t. t ∈ K s} (d s) ∂D) ∂Π⇩M i∈UNIV. K i)"
    apply (rule nn_integral_cong_AE)
    apply (subst walk.ctr)
    apply (simp cong del: if_weak_cong)
    apply (intro UNIV_I AE_component)
    apply (auto simp: AE_measure_pmf_iff)
    done
  also have "… = (∫⇧+d. ∫⇧+ω. f (d s ## ω) * indicator (K s) (d s) ∂T (d s) ∂Π⇩M i∈UNIV. K i)"
    by (subst T) (simp_all split: split_indicator)
  also have "… = (∫⇧+t. ∫⇧+ω. f (t ## ω) * indicator (K s) t ∂T t ∂K s)"
    by (subst (2) PiM_component[symmetric]) (simp_all add: nn_integral_distr)
  also have "… = (∫⇧+t. ∫⇧+ω. f (t ## ω) ∂T t ∂K s)"
    by (rule nn_integral_cong_AE) (simp add: AE_measure_pmf_iff)
  finally show ?thesis .
qed

lemma nn_integral_T_gfp:
  fixes g
  defines "l ≡ λf ω. g (shd ω) (f (stl ω))"
  assumes [measurable]: "case_prod g ∈ borel_measurable (count_space UNIV ⨂⇩M borel)"
  assumes cont_g[THEN inf_continuous_compose, order_continuous_intros]: "⋀s. inf_continuous (g s)"
  assumes int_g: "⋀f s. f ∈ borel_measurable S ⟹ (∫⇧+ω. g s (f ω) ∂T s) = g s (∫⇧+ω. f ω ∂T s)"
  assumes bnd_g: "⋀f s. g s f ≤ b" "0 ≤ b" "b < ∞"
  shows "(∫⇧+ω. gfp l ω ∂T s) = gfp (λf s. ∫⇧+t. g t (f t) ∂K s) s"
proof (rule nn_integral_gfp)
  show "⋀s. sets (T s) = sets S" "⋀F. F ∈ borel_measurable S ⟹ l F ∈ borel_measurable S"
    by (auto simp: l_def)
  show "⋀s. emeasure (T s) (space (T s)) ≠ 0"
   by (rewrite T.emeasure_space_1) simp
  { fix s F
    have "integral⇧N (T s) (l F) ≤ (∫⇧+x. b ∂T s)"
      by (intro nn_integral_mono) (simp add: l_def bnd_g)
    also have "… < ∞"
      using bnd_g by simp
    finally show "integral⇧N (T s) (l F) < ∞" . }
  show "inf_continuous (λf s. ∫⇧+ t. g t (f t) ∂K s)"
  proof (intro order_continuous_intros)
    fix f s
    have "(∫⇧+ t. g t (f t) ∂K s) ≤ (∫⇧+ t. b ∂K s)"
      by (intro nn_integral_mono bnd_g)
    also have "… < ∞"
      using bnd_g by simp
    finally show "(∫⇧+ t. g t (f t) ∂K s) ≠ ∞"
      by simp
  qed simp
next
  fix s and F :: "'s stream ⇒ ennreal" assume "F ∈ borel_measurable S"
  then show "integral⇧N (T s) (l F) = (∫⇧+ t. g t (integral⇧N (T t) F) ∂K s) "
    by (rewrite nn_integral_T) (simp_all add: l_def int_g)
qed (auto intro!: order_continuous_intros simp: l_def)

lemma nn_integral_T_lfp:
  fixes g
  defines "l ≡ λf ω. g (shd ω) (f (stl ω))"
  assumes [measurable]: "case_prod g ∈ borel_measurable (count_space UNIV ⨂⇩M borel)"
  assumes cont_g[THEN sup_continuous_compose, order_continuous_intros]: "⋀s. sup_continuous (g s)"
  assumes int_g: "⋀f s. f ∈ borel_measurable S ⟹ (∫⇧+ω. g s (f ω) ∂T s) = g s (∫⇧+ω. f ω ∂T s)"
  shows "(∫⇧+ω. lfp l ω ∂T s) = lfp (λf s. ∫⇧+t. g t (f t) ∂K s) s"
proof (rule nn_integral_lfp)
  show "⋀s. sets (T s) = sets S" "⋀F. F ∈ borel_measurable S ⟹ l F ∈ borel_measurable S"
    by (auto simp: l_def)
next
  fix s and F :: "'s stream ⇒ ennreal" assume "F ∈ borel_measurable S"
  then show "integral⇧N (T s) (l F) = (∫⇧+ t. g t (integral⇧N (T t) F) ∂K s) "
    by (rewrite nn_integral_T) (simp_all add: l_def int_g)
qed (auto simp: l_def intro!: order_continuous_intros)

lemma emeasure_Collect_T:
  assumes f[measurable]: "Measurable.pred S P"
  shows "emeasure (T s) {x∈space (T s). P x} = (∫⇧+t. emeasure (T t) {x∈space (T t). P (t ## x)} ∂K s)"
  apply (subst (1 2) nn_integral_indicator[symmetric])
  apply simp
  apply simp
  apply (subst nn_integral_T)
  apply (auto intro!: nn_integral_cong simp add: space_stream_space indicator_def)
  done

lemma AE_T_iff:
  assumes [measurable]: "Measurable.pred S P"
  shows "(AE ω in T x. P ω) ⟷ (∀y∈K x. AE ω in T y. P (y ## ω))"
  by (simp add: AE_iff_nn_integral nn_integral_T[where s=x])
     (auto simp add: nn_integral_0_iff_AE AE_measure_pmf_iff split: split_indicator)

lemma AE_T_alw:
  assumes [measurable]: "Measurable.pred S P"
  assumes P: "⋀s. (x, s) ∈ acc ⟹ AE ω in T s. P ω"
  shows "AE ω in T x. alw P ω"
proof -
  define F where "F = (λp x. P x ∧ p (stl x))"
  have [measurable]: "⋀p. Measurable.pred S p ⟹ Measurable.pred S (F p)"
    by (auto simp: F_def)
  have "almost_everywhere (T s) ((F ^^ i) top)"
    if "(x, s) ∈ acc" for i s
    using that
  proof (induction i arbitrary: s)
    case (Suc i) then show ?case
      apply simp
      apply (subst F_def)
      apply (simp add: P)
      apply (subst AE_T_iff)
      apply (measurable; simp)
      apply (auto dest: rtrancl_into_rtrancl)
      done
  qed simp
  then have "almost_everywhere (T x) (gfp F)"
    by (subst inf_continuous_gfp) (auto simp: inf_continuous_def AE_all_countable F_def)
  then show ?thesis
    by (simp add: alw_def F_def)
qed

lemma emeasure_suntil_disj:
  assumes [measurable]: "Measurable.pred S P"
  assumes *: "⋀t. AE ω in T t. ¬ (P ⊓ (HLD X ⊓ nxt (HLD X suntil P))) ω"
  shows "emeasure (T s) {ω∈space (T s). (HLD X suntil P) ω} =
    lfp (λF s. emeasure (T s) {ω∈space (T s). P ω} + (∫⇧+t. F t * indicator X t ∂K s)) s"
  unfolding suntil_lfp
proof (rule emeasure_lfp[where s=s])
  fix F t assume [measurable]: "Measurable.pred (T s) F" and
    F: "F ≤ lfp (λa b. P b ∨ HLD X b ∧ a (stl b))"
  have "emeasure (T t) {ω ∈ space (T s). P ω ∨ HLD X ω ∧ F (stl ω)} =
    emeasure (T t) {ω ∈ space (T t). P ω} + emeasure (T t) {ω∈space (T t). HLD X ω ∧ F (stl ω)}"
  proof (rule emeasure_add_AE)
    show "AE x in T t. ¬ (x ∈ {ω ∈ space (T t). P ω} ∧ x ∈ {ω ∈ space (T t). HLD X ω ∧ F (stl ω)})"
      using * by eventually_elim (insert F, auto simp: suntil_lfp[symmetric])
  qed auto
  also have "emeasure (T t) {ω∈space (T t). HLD X ω ∧ F (stl ω)} =
    (∫⇧+t. emeasure (T t) {ω ∈ space (T s). F ω} * indicator X t ∂K t)"
    by (subst emeasure_Collect_T) (auto intro!: nn_integral_cong split: split_indicator)
  finally show "emeasure (T t) {ω ∈ space (T s). P ω ∨ HLD X ω ∧ F (stl ω)} =
    emeasure (T t) {ω ∈ space (T t). P ω} + (∫⇧+ t. emeasure (T t) {ω ∈ space (T s). F ω} * indicator X t ∂K t)" .
qed (auto intro!: order_continuous_intros split: split_indicator)

lemma emeasure_HLD_nxt:
  assumes [measurable]: "Measurable.pred S P"
  shows "emeasure (T s) {ω∈space (T s). (X ⋅ P) ω} =
    (∫⇧+x. emeasure (T x) {ω∈space (T x). P ω} * indicator X x ∂K s)"
  by (subst emeasure_Collect_T)
     (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff split: split_indicator)

lemma emeasure_HLD:
  "emeasure (T s) {ω∈space (T s). HLD X ω} = emeasure (K s) X"
  using emeasure_HLD_nxt[of "λω. True" s X] T.emeasure_space_1 by simp

lemma emeasure_suntil_HLD:
  assumes [measurable]: "Measurable.pred S P"
  shows "emeasure (T s) {x∈space (T s). (not (HLD {t}) suntil (HLD {t} aand nxt P)) x} =
   emeasure (T s) {x∈space (T s). ev (HLD {t}) x} * emeasure (T t) {x∈space (T t). P x}"
proof -
  let ?P = "emeasure (T t) {ω∈space (T t). P ω}"
  let ?F = "λQ F s. emeasure (T s) {ω∈space (T s). Q ω} + (∫⇧+t'. F t' * indicator (- {t}) t' ∂K s)"
  have "emeasure (T s) {x∈space (T s). (HLD (-{t}) suntil ({t} ⋅ P)) x} = lfp (?F ({t} ⋅ P)) s"
    by (rule emeasure_suntil_disj) (auto simp: HLD_iff)
  also have "lfp (?F ({t} ⋅ P)) = (λs. lfp (?F (HLD {t})) s * ?P)"
  proof (rule lfp_transfer[symmetric, where α="λx s. x s * emeasure (T t) {ω∈space (T t). P ω}"])
    fix F show "(λs. ?F (HLD {t}) F s * ?P) = ?F ({t} ⋅ P) (λs. F s * ?P)"
      unfolding emeasure_HLD emeasure_HLD_nxt[OF assms] distrib_right
      by (auto simp: fun_eq_iff nn_integral_multc[symmetric]
               intro!: arg_cong2[where f="(+)"] nn_integral_cong ac_simps
               split: split_indicator)
  qed (auto intro!: order_continuous_intros sup_continuous_mono lfp_upperbound
            intro: le_funI add_nonneg_nonneg
            simp: bot_ennreal split: split_indicator)
  also have "lfp (?F (HLD {t})) s = emeasure (T s) {x∈space (T s). (HLD (-{t}) suntil HLD {t}) x}"
    by (rule emeasure_suntil_disj[symmetric]) (auto simp: HLD_iff)
  finally show ?thesis
    by (simp add: HLD_iff[abs_def] ev_eq_suntil)
qed

lemma AE_suntil:
  assumes [measurable]: "Measurable.pred S P"
  shows "(AE x in T s. (not (HLD {t}) suntil (HLD {t} aand nxt P)) x) ⟷
   (AE x in T s. ev (HLD {t}) x) ∧ (AE x in T t. P x)"
  apply (subst (1 2 3) T.prob_Collect_eq_1[symmetric])
  apply simp
  apply simp
  apply simp
  apply (simp_all add: measure_def emeasure_suntil_HLD del: space_T nxt.simps)
  apply (auto simp: T.emeasure_eq_measure mult_eq_1)
  done

subsection ‹Fairness›

definition fair :: "'s ⇒ 's ⇒ 's stream ⇒ bool" where
  "fair s t = alw (ev (HLD {s})) impl alw (ev (HLD {s} aand nxt (HLD {t})))"

lemma AE_T_fair:
  assumes "t' ∈ K t"
  shows "AE ω in T s. fair t t' ω"
proof -
  let ?M = "λP s. emeasure (T s) {ω∈space (T s). P ω}"
  let ?t = "HLD {t}" and ?t' = "HLD {t'}"
  define N where "N = alw (ev ?t) aand alw (not (?t aand nxt ?t'))"
  let ?until = "not ?t suntil (?t aand nxt (not ?t' aand nxt N))"
  have N_stl: "⋀ω. N ω ⟹ N (stl ω)"
    by (auto simp: N_def)
  have [measurable]: "Measurable.pred S N"
    unfolding N_def by measurable

  let ?c = "pmf (K t) t'"
  let ?R = "λx. 1 ⊓ x * (1 - ennreal ?c)"
  have "mono ?R"
    by (intro monoI mult_right_mono inf_mono) (auto simp: mono_def field_simps )
  have "⋀s. ?M N s ≤ gfp ?R"
  proof (induction rule: gfp_ordinal_induct[OF ‹mono ?R›])
    fix x s assume x: "⋀s. ?M N s ≤ x"
    { fix ω assume "N ω"
      then have "ev (HLD {t}) ω" "N ω"
        by (auto simp: N_def)
      then have "?until ω"
        by (induct rule: ev_induct_strong) (auto simp: N_def intro: suntil.intros dest: N_stl) }
    then have "?M N s ≤ ?M ?until s"
      by (intro emeasure_mono_AE) auto
    also have "… = ?M (ev ?t) s * ?M (not ?t' aand nxt N) t"
      by (simp_all add: emeasure_suntil_HLD del: nxt.simps space_T)
    also have "… ≤ ?M (ev ?t) s * (∫⇧+s'. 1 ⊓ x * indicator (UNIV - {t'}) s' ∂K t)"
      by (auto intro!: mult_left_mono nn_integral_mono T.measure_le_1 emeasure_mono
               split: split_indicator simp add: x emeasure_Collect_T[of _ t] simp del: space_T)
    also have "… ≤ 1 * (∫⇧+s'. 1 ⊓ x * indicator (UNIV - {t'}) s' ∂K t)"
      by (intro mult_right_mono T.measure_le_1) simp
    finally show "?M N s ≤ 1 ⊓ x * (1 - ennreal ?c)"
      by (subst (asm) nn_integral_cmult_indicator) (auto simp: emeasure_Diff emeasure_pmf_single)
  qed (auto intro: Inf_greatest)
  also
  from ‹mono ?R› have "gfp ?R = ?R (gfp ?R)" by (rule gfp_unfold)
  then have "gfp ?R ≤ ?R (gfp ?R)" by simp
  with assms[THEN pmf_positive] have "gfp ?R ≤ 0"
    by (cases "gfp ?R")
       (auto simp: top_unique inf_ennreal.rep_eq field_simps mult_le_0_iff ennreal_1[symmetric]
                   pmf_le_1 ennreal_minus ennreal_mult[symmetric] ennreal_le_iff2 inf_min min_def
             simp del: ennreal_1
             split: if_split_asm)
  finally have "⋀s. AE ω in T s. ¬ N ω"
    by (subst AE_iff_measurable[OF _ refl]) (auto intro: antisym simp: le_fun_def)
  then have "AE ω in T s. alw (not N) ω"
    by (intro AE_T_alw) auto
  moreover
  { fix ω assume "alw (ev (HLD {t})) ω"
    then have "alw (alw (ev (HLD {t}))) ω"
      unfolding alw_alw .
    moreover assume "alw (not N) ω"
    then have "alw (alw (ev (HLD {t})) impl ev (HLD {t} aand nxt (HLD {t'}))) ω"
      unfolding N_def not_alw_iff not_ev_iff de_Morgan_disj de_Morgan_conj not_not imp_conv_disj .
    ultimately have "alw (ev (HLD {t} aand nxt (HLD {t'}))) ω"
      by (rule alw_mp) }
  then have "∀ω. alw (not N) ω ⟶ fair t t' ω"
    by (auto simp: fair_def)
  ultimately show ?thesis
    by (simp add: eventually_mono)
qed

lemma enabled_imp_trancl:
  assumes "alw (HLD B) ω" "enabled s ω"
  shows "alw (HLD (acc_on B `` {s})) ω"
proof -
  define t where "t = s"
  then have "(s, t) ∈ acc_on B"
    by auto
  moreover note ‹alw (HLD B) ω›
  moreover note ‹enabled s ω›[unfolded ‹t == s›[symmetric]]
  ultimately show ?thesis
  proof (coinduction arbitrary: t ω rule: alw_coinduct)
    case stl from this(1,2,3) show ?case
      by (auto simp: enabled.simps[of _ ω] alw.simps[of _ ω] HLD_iff
                 intro!: exI[of _ "shd ω"] rtrancl_trans[of s t])
  next
    case (alw t ω) then show ?case
     by (auto simp: HLD_iff enabled.simps[of _ ω] alw.simps[of _ ω] intro!: rtrancl_trans[of s t])
  qed
qed

lemma AE_T_reachable: "AE ω in T s. alw (HLD (acc `` {s})) ω"
  using AE_T_enabled
proof eventually_elim
  fix ω assume "enabled s ω"
  from enabled_imp_trancl[of UNIV, OF _ this]
  show "alw (HLD (acc `` {s})) ω"
    by (auto simp: HLD_iff[abs_def] all_imp_alw)
qed

lemma AE_T_all_fair: "AE ω in T s. ∀(t,t')∈SIGMA t:UNIV. K t. fair t t' ω"
proof -
  let ?Rn = "SIGMA s:(acc `` {s}). K s"
  have "AE ω in T s. ∀(t,t')∈?Rn. fair t t' ω"
  proof (subst AE_ball_countable)
    show "countable ?Rn"
      by (intro countable_SIGMA countable_rtrancl[OF countable_Image]) (auto simp: Image_def)
  qed (auto intro!: AE_T_fair)
  then show ?thesis
    using AE_T_reachable
  proof (eventually_elim, safe)
    fix ω t t' assume "∀(t,t')∈?Rn. fair t t' ω" "t' ∈ K t" and alw: "alw (HLD (acc `` {s})) ω"
    moreover
    { assume "t ∉ acc `` {s}"
      then have "alw (not (HLD {t})) ω"
        by (intro alw_mono[OF alw]) (auto simp: HLD_iff)
      then have "not (alw (ev (HLD {t}))) ω"
        unfolding not_alw_iff not_ev_iff by auto
      then have "fair t t' ω"
        unfolding fair_def by auto }
    ultimately show "fair t t' ω"
      by auto
  qed
qed

lemma fair_imp: assumes "fair t t' ω" "alw (ev (HLD {t})) ω" shows "alw (ev (HLD {t'})) ω"
proof -
  { fix ω assume "ev (HLD {t} aand nxt (HLD {t'})) ω" then have "ev (HLD {t'}) ω"
      by induction auto }
  with assms show ?thesis
    by (auto simp: fair_def elim!: alw_mp intro: all_imp_alw)
qed

lemma AE_T_ev_HLD:
  assumes exiting: "⋀t. (s, t) ∈ acc_on (-B) ⟹ ∃t'∈B. (t, t') ∈ acc"
  assumes fin: "finite (acc_on (-B) `` {s})"
  shows "AE ω in T s. ev (HLD B) ω"
  using AE_T_all_fair AE_T_enabled
proof eventually_elim
  fix ω assume fair: "∀(t, t')∈(SIGMA s:UNIV. K s). fair t t' ω" and "enabled s ω"

  show "ev (HLD B) ω"
  proof (rule ccontr)
    assume "¬ ev (HLD B) ω"
    then have "alw (HLD (- B)) ω"
      by (simp add: not_ev_iff HLD_iff[abs_def])
    from enabled_imp_trancl[OF this ‹enabled s ω›]
    have "alw (HLD (acc_on (-B) `` {s})) ω"
      by (simp add: Diff_eq)
    from pigeonhole_stream[OF this fin]
    obtain t where "(s, t) ∈ acc_on (-B)" "alw (ev (HLD {t})) ω"
      by auto
    from exiting[OF this(1)] obtain t' where "(t, t') ∈ acc" "t' ∈ B"
      by auto
    from this(1) have "alw (ev (HLD {t'})) ω"
    proof induction
      case (step u w) then show ?case
        using fair fair_imp[of u w ω] by auto
    qed fact
    { assume "ev (HLD {t'}) ω" then have "ev (HLD B) ω"
      by (rule ev_mono) (auto simp: HLD_iff ‹t' ∈ B›) }
    then show False
      using ‹alw (ev (HLD {t'})) ω› ‹¬ ev (HLD B) ω› by auto
  qed
qed

lemma AE_T_ev_HLD':
  assumes exiting: "⋀s. s ∉ X ⟹ ∃t∈X. (s, t) ∈ acc"
  assumes fin: "finite (-X)"
  shows "AE ω in T s. ev (HLD X) ω"
proof (rule AE_T_ev_HLD)
  show "⋀t. (s, t) ∈ acc_on (- X) ⟹ ∃t'∈X. (t, t') ∈ acc"
    using exiting by (auto elim: rtrancl.cases)
  have "acc_on (- X) `` {s} ⊆ -X ∪ {s}"
    by (auto elim: rtrancl.cases)
  with fin show "finite (acc_on (- X) `` {s})"
    by (auto dest: finite_subset )
qed

lemma AE_T_max_sfirst:
  assumes [measurable]: "Measurable.pred S X"
  assumes AE: "AE ω in T c. sfirst X (c ## ω) < ∞" and "0 < e"
  shows "∃N::nat. 𝒫(ω in T c. N < sfirst X (c ## ω)) < e" (is "∃N. ?P N < e")
proof -
  have "?P ⇢ measure (T c) (⋂N::nat. {bT ∈ space (T c). N < sfirst X (c ## bT)})"
    using dual_order.strict_trans enat_ord_simps(2)
    by (intro T.finite_Lim_measure_decseq) (force simp: decseq_Suc_iff simp del: enat_ord_simps)+
  also have "measure (T c) (⋂N::nat. {bT ∈ space (T c). N < sfirst X (c ## bT)}) =
      𝒫(bT in T c. sfirst X (c ## bT) = ∞)"
    by (auto simp del: not_infinity_eq intro!: arg_cong[where f="measure (T c)"])
       (metis less_irrefl not_infinity_eq)
  also have "𝒫(bT in T c. sfirst X (c ## bT) = ∞) = 0"
    using AE by (intro T.prob_eq_0_AE) auto
  finally have "∃N. ∀n≥N. norm (?P n - 0) < e"
    using ‹0 < e› by (rule LIMSEQ_D)
  then show ?thesis
    by (auto simp: measure_nonneg)
qed

subsection ‹First Hitting Time›

lemma nn_integral_sfirst_finite':
  assumes "s ∉ H"
  assumes [simp]: "finite (acc_on (-H) `` {s})"
  assumes until: "AE ω in T s. ev (HLD H) ω"
  shows "(∫⇧+ ω. sfirst (HLD H) ω ∂T s) ≠ ∞"
proof -
  have R_ne[simp]: "acc_on (-H) `` {s} ≠ {}"
    by auto
  have [measurable]: "H ∈ sets (count_space UNIV)"
    by simp

  let ?Pf = "λn t. 𝒫(ω in T t. enat n < sfirst (HLD H) (t ## ω))"
  have Pf_mono: "⋀N n t. N ≤ n ⟹ ?Pf n t ≤ ?Pf N t"
    by (auto intro!: T.finite_measure_mono simp del: enat_ord_code(1) simp: enat_ord_code(1)[symmetric])

  have not_H: "⋀t. (s, t) ∈ acc_on (-H) ⟹ t ∉ H"
    using ‹s ∉ H› by (auto elim: rtrancl.cases)

  have "∀⇩F n in sequentially. ∀t∈acc_on (-H)``{s}. ?Pf n t < 1"
  proof (safe intro!: eventually_ball_finite)
    fix t assume "(s, t) ∈ acc_on (-H)"
    then have "AE ω in T t. sfirst (HLD H) (t ## ω) < ∞"
      unfolding sfirst_finite
    proof induction
      case (step t u) with step.IH show ?case
        by (subst (asm) AE_T_iff) (auto simp: ev_Stream not_H)
    qed (simp add: ev_Stream eventually_frequently_simps until)
    from AE_T_max_sfirst[OF _ this, of 1]
    obtain N where "?Pf N t < 1" by auto
    with Pf_mono[of N] show "∀⇩F n in sequentially. ?Pf n t < 1"
      by (auto simp: eventually_sequentially intro: le_less_trans)
  qed simp
  then obtain n where "⋀t. (s, t) ∈ acc_on (-H) ⟹ ?Pf n t < 1"
    by (auto simp: eventually_sequentially)
  moreover define d where "d = Max (?Pf n ` acc_on (-H) `` {s})"
  ultimately have d: "0 ≤ d" "d < 1" "⋀t. (s, t) ∈ acc_on (-H) ⟹ ?Pf (Suc n) t ≤ d"
    using Pf_mono[of n "Suc n"] by (auto simp: Max_ge_iff measure_nonneg)

  let ?F = "λF ω. if shd ω ∈ H then 0 else F (stl ω) + 1 :: ennreal"
  have "sup_continuous ?F"
    by (intro order_continuous_intros)
  then have "mono ?F"
    by (rule sup_continuous_mono)
  have lfp_nonneg[simp]: "⋀ω. 0 ≤ lfp ?F ω"
    by (subst lfp_unfold[OF ‹mono ?F›]) auto

  let ?I = "λF s. ∫⇧+t. (if t ∈ H then 0 else F t + 1) ∂K s"
  have "sup_continuous ?I"
    by (intro order_continuous_intros) auto
  then have "mono ?I"
    by (rule sup_continuous_mono)

  define p where "p = Suc n / (1 - d)"
  have p: "p = Suc n + d * p"
    unfolding p_def using d(1,2) by (auto simp: field_simps)
  have [simp]: "0 ≤ p"
    using d(1,2) by (auto simp: p_def)

  have "(∫⇧+ ω. sfirst (HLD H) ω ∂T s) = (∫⇧+ ω. lfp ?F ω ∂T s)"
  proof (intro nn_integral_cong_AE)
    show "AE x in T s. sfirst (HLD H) x = lfp ?F x"
      using until
    proof eventually_elim
      fix ω assume "ev (HLD H) ω" then show "sfirst (HLD H) ω = lfp ?F ω"
        by (induction rule: ev_induct_strong;
            subst lfp_unfold[OF ‹mono ?F›], simp add: HLD_iff[abs_def] ac_simps max_absorb2)
    qed
  qed
  also have "… = lfp (?I^^Suc n) s"
    unfolding lfp_funpow[OF ‹mono ?I›]
    by (subst nn_integral_T_lfp)
       (auto simp: nn_integral_add max_absorb2 intro!: order_continuous_intros)
  also have "lfp (?I^^Suc n) t ≤ p" if "(s, t) ∈ acc_on (-H)" for t
    using that
  proof (induction arbitrary: t rule: lfp_ordinal_induct[of "?I^^Suc n"])
    case (step S)
    have "(?I^^i) S t ≤ i + ?Pf i t * ennreal p" for i
      using step(3)
    proof (induction i arbitrary: t)
      case 0 then show ?case
        using T.prob_space step(1)
        by (auto simp add: zero_ennreal_def[symmetric] not_H zero_enat_def[symmetric] one_ennreal_def[symmetric])
    next
      case (Suc i)
      then have "t ∉ H"
        by (auto simp: not_H)
      from Suc.prems have "⋀t'. t' ∈ K t ⟹ t' ∉ H ⟹ (s, t') ∈ acc_on (-H)"
        by (rule rtrancl_into_rtrancl) (insert Suc.prems, auto dest: not_H)
      then have "(?I ^^ Suc i) S t ≤ ?I (λt. i + ennreal (?Pf i t) * p) t"
        by (auto simp: AE_measure_pmf_iff simp del: sfirst_eSuc space_T
                 intro!: nn_integral_mono_AE add_mono max.mono Suc)
      also have "… ≤ (∫⇧+ t. ennreal (Suc i) + ennreal 𝒫(ω in T t. enat i < sfirst (HLD H) (t ## ω)) * p ∂K t)"
        by (intro nn_integral_mono) auto
      also have "… ≤ Suc i + ennreal (?Pf (Suc i) t) * p"
        unfolding T.emeasure_eq_measure[symmetric]
        by (subst (2) emeasure_Collect_T)
           (auto simp: ‹t ∉ H› eSuc_enat[symmetric] nn_integral_add nn_integral_multc ennreal_of_nat_eq_real_of_nat)
      finally show ?case
        by (simp add: ennreal_of_nat_eq_real_of_nat)
    qed
    then have "(?I^^Suc n) S t ≤ Suc n + ?Pf (Suc n) t * ennreal p" .
    also have "… ≤ p"
      using d step by (subst (2) p) (auto intro!: mult_right_mono simp: ennreal_of_nat_eq_real_of_nat ennreal_mult)
    finally show ?case .
  qed (auto simp: SUP_least intro!: mono_pow ‹mono ?I› simp del: funpow.simps)
  finally show ?thesis
    unfolding p_def by (auto simp: top_unique)
qed

lemma nn_integral_sfirst_finite:
  assumes [simp]: "finite (acc_on (-H) `` {s})"
  assumes until: "AE ω in T s. ev (HLD H) ω"
  shows "(∫⇧+ ω. sfirst (HLD H) (s ## ω) ∂T s) ≠ ∞"
proof cases
  assume "s ∉ H" then show ?thesis
    using nn_integral_sfirst_finite'[of s H] until by (simp add: nn_integral_add)
qed (simp add: sfirst.simps)

lemma prob_T:
  assumes P: "Measurable.pred S P"
  shows "𝒫(ω in T s. P ω) = (∫t. 𝒫(ω in T t. P (t ## ω)) ∂K s)"
  using emeasure_Collect_T[OF P, of s] unfolding T.emeasure_eq_measure
  by (subst (asm) nn_integral_eq_integral)
     (auto intro!: measure_pmf.integrable_const_bound[where B=1])

lemma T_subprob[measurable]: "T ∈ measurable (measure_pmf I) (subprob_algebra S)"
  by (auto intro!: space_bind simp: space_subprob_algebra) unfold_locales

subsection ‹Markov chain with Initial Distribution›

definition T' :: "'s pmf ⇒ 's stream measure" where
  "T' I = bind I (λs. distr (T s) S ((##) s))"

lemma distr_Stream_subprob:
  "(λs. distr (T s) S ((##) s)) ∈ measurable (measure_pmf I) (subprob_algebra S)"
  apply (intro measurable_distr2[OF _ T_subprob])
  apply (subst measurable_cong_sets[where M'="count_space UNIV ⨂⇩M S" and N'=S])
  apply (rule sets_pair_measure_cong)
  apply auto
  done

lemma sets_T': "sets (T' I) = sets S"
  by (simp add: T'_def)

lemma prob_space_T': "prob_space (T' I)"
  unfolding T'_def
proof (rule measure_pmf.prob_space_bind)
  show "AE s in I. prob_space (distr (T s) S ((##) s))"
    by (intro AE_measure_pmf_iff[THEN iffD2] ballI T.prob_space_distr) simp
qed (rule distr_Stream_subprob)

lemma AE_T':
  assumes [measurable]: "Measurable.pred S P"
  shows "(AE x in T' I. P x) ⟷ (∀s∈I. AE x in T s. P (s ## x))"
  unfolding T'_def by (simp add: AE_bind[OF distr_Stream_subprob] AE_measure_pmf_iff AE_distr_iff)

lemma emeasure_T':
  assumes [measurable]: "X ∈ sets S"
  shows "emeasure (T' I) X = (∫⇧+s. emeasure (T s) {ω∈space S. s ## ω ∈ X} ∂I)"
  unfolding T'_def
  by (simp add: emeasure_bind[OF _ distr_Stream_subprob] emeasure_distr vimage_def Int_def conj_ac)

lemma prob_T':
  assumes [measurable]: "Measurable.pred S P"
  shows "𝒫(x in T' I. P x) = (∫s. 𝒫(x in T s. P (s ## x)) ∂I)"
proof -
  interpret T': prob_space "T' I" by (rule prob_space_T')
  show ?thesis
    using emeasure_T'[of "{x∈space (T' I). P x}" I]
    unfolding T'.emeasure_eq_measure T.emeasure_eq_measure sets_eq_imp_space_eq[OF sets_T']
    apply simp
    apply (subst (asm) nn_integral_eq_integral)
    apply (auto intro!: measure_pmf.integrable_const_bound[where B=1] integral_cong arg_cong2[where f=measure]
                simp: AE_measure_pmf measure_nonneg space_stream_space)
    done
qed

lemma T_eq_T': "T s = T' (K s)"
proof (rule measure_eqI)
  fix X assume X: "X ∈ sets (T s)"
  then have [measurable]: "X ∈ sets S"
    by simp
  have X_eq: "X = {x∈space (T s). x ∈ X}"
    using sets.sets_into_space[OF X] by auto
  show "emeasure (T s) X = emeasure (T' (K s)) X"
    apply (subst X_eq)
    apply (subst emeasure_Collect_T, simp)
    apply (subst emeasure_T', simp)
    apply simp
    done
qed (simp add: sets_T')

lemma T_eq_bind: "T s = (measure_pmf (K s) ⤜ (λt. distr (T t) S ((##) t)))"
  by (subst T_eq_T') (simp add: T'_def)

lemma T_split:
  "T s = (T s ⤜ (λω. distr (T ((s ## ω) !! n)) S (λω'. stake n ω @- ω')))"
proof (induction n arbitrary: s)
  case 0 then show ?case
    apply (simp add: distr_cong[OF refl sets_T[symmetric, of s] refl])
    apply (subst bind_const')
    apply unfold_locales
    ..
next
  case (Suc n)
  let ?K = "measure_pmf (K s)" and ?m = "λn ω ω'. stake n ω @- ω'"
  note sets_stream_space_cong[simp, measurable_cong]

  have "T s = (?K ⤜ (λt. distr (T t) S ((##) t)))"
    by (rule T_eq_bind)
  also have "… = (?K ⤜ (λt. distr (T t ⤜ (λω. distr (T ((t ## ω) !! n)) S (?m n ω))) S ((##) t)))"
    unfolding Suc[symmetric] ..
  also have "… = (?K ⤜ (λt. T t ⤜ (λω. distr (distr (T ((t ## ω) !! n)) S (?m n ω)) S ((##) t))))"
    by (simp add: distr_bind[where K=S, OF measurable_distr2[where M=S]] space_stream_space)
  also have "… = (?K ⤜ (λt. T t ⤜ (λω. distr (T ((t ## ω) !! n)) S (?m (Suc n) (t ## ω)))))"
    by (simp add: distr_distr space_stream_space comp_def)
  also have "… = (?K ⤜ (λt. distr (T t) S ((##) t) ⤜ (λω. distr (T (ω !! n)) S (?m (Suc n) ω))))"
    by (simp add: space_stream_space bind_distr[OF _ measurable_distr2[where M=S]] del: stake.simps)
  also have "… = (T s ⤜ (λω. distr (T (ω !! n)) S (?m (Suc n) ω)))"
    unfolding T_eq_bind[of s]
    by (subst bind_assoc[OF measurable_distr2[where M=S] measurable_distr2[where M=S], OF _ T_subprob])
       (simp_all add: space_stream_space del: stake.simps)
  finally show ?case
    by simp
qed

lemma nn_integral_T_split:
  assumes f[measurable]: "f ∈ borel_measurable S"
  shows "(∫⇧+ω. f ω ∂T s) = (∫⇧+ω. (∫⇧+ω'. f (stake n ω @- ω') ∂T ((s ## ω) !! n)) ∂T s)"
  apply (subst T_split[of s n])
  apply (simp add: nn_integral_bind[OF f measurable_distr2[where M=S]])
  apply (subst nn_integral_distr)
  apply (simp_all add: space_stream_space)
  done

lemma emeasure_T_split:
  assumes P[measurable]: "Measurable.pred S P"
  shows "emeasure (T s) {ω∈space (T s). P ω} =
      (∫⇧+ω. emeasure (T ((s ## ω) !! n)) {ω'∈space (T ((s ## ω) !! n)). P (stake n ω @- ω')} ∂T s)"
  apply (subst T_split[of s n])
  apply (subst emeasure_bind[OF _ measurable_distr2[where M=S]])
  apply (simp_all add: )
  apply (simp add: space_stream_space)
  apply (subst emeasure_distr)
  apply simp_all
  apply (simp_all add: space_stream_space)
  done

lemma prob_T_split:
  assumes P[measurable]: "Measurable.pred S P"
  shows "𝒫(ω in T s. P ω) = (∫ω. 𝒫(ω' in T ((s ## ω) !! n). P (stake n ω @- ω')) ∂T s)"
  using emeasure_T_split[OF P, of s n]
  unfolding T.emeasure_eq_measure
  by (subst (asm) nn_integral_eq_integral)
     (auto intro!: T.integrable_const_bound[where B=1] measure_measurable_subprob_algebra2[where N=S]
           simp: T.emeasure_eq_measure SIGMA_Collect_eq)

lemma enabled_imp_alw:
  "(⋃s∈X. set_pmf (K s)) ⊆ X ⟹ x ∈ X ⟹ enabled x ω ⟹ alw (HLD X) ω"
proof (coinduction arbitrary: ω x)
  case alw then show ?case
    unfolding enabled.simps[of _ ω]
    by (auto simp: HLD_iff)
qed

lemma alw_HLD_iff_sconst:
  "alw (HLD {x}) ω ⟷ ω = sconst x"
proof
  assume "alw (HLD {x}) ω" then show "ω = sconst x"
    by (coinduction arbitrary: ω) (auto simp: HLD_iff)
qed (auto simp: alw_sconst HLD_iff)

lemma enabled_iff_sconst:
  assumes [simp]: "set_pmf (K x) = {x}" shows "enabled x ω ⟷ ω = sconst x"
proof
  assume "enabled x ω" then show "ω = sconst x"
    by (coinduction arbitrary: ω) (auto elim: enabled.cases)
next
  assume "ω = sconst x" then show "enabled x ω"
    by (coinduction arbitrary: ω) auto
qed

lemma AE_sconst:
  assumes [simp]: "set_pmf (K x) = {x}"
  shows "(AE ω in T x. P ω) ⟷ P (sconst x)"
proof -
  have "(AE ω in T x. P ω) ⟷ (AE ω in T x. P ω ∧ ω = sconst x)"
    using AE_T_enabled[of x] by (simp add: enabled_iff_sconst)
  also have "… = (AE ω in T x. P (sconst x) ∧ ω = sconst x)"
    by (simp del: AE_conj_iff cong: rev_conj_cong)
  also have "… = (AE ω in T x. P (sconst x))"
    using AE_T_enabled[of x] by (simp add: enabled_iff_sconst)
  finally show ?thesis
    by simp
qed

lemma ev_eq_lfp: "ev P = lfp (λF ω. P ω ∨ (¬ P ω ∧ F (stl ω)))"
  unfolding ev_def by (intro antisym lfp_mono) blast+

lemma INF_eq_zero_iff_ennreal: "((⨅i∈A. f i) = (0::ennreal)) = (∀x>0. ∃i∈A. f i < x)"
  using INF_eq_bot_iff[where 'a=ennreal] unfolding bot_ennreal_def zero_ennreal_def by auto

lemma inf_continuous_cmul: 
  fixes c :: ennreal
  assumes f: "inf_continuous f" and c: "c < ⊤" 
  shows "inf_continuous (λx. c * f x)"
proof (rule inf_continuous_compose[OF _ f], clarsimp simp add: inf_continuous_def)
  fix M :: "nat ⇒ ennreal" assume M: "decseq M" 
  show "c * (⨅i. M i) = (⨅i. c * M i)"
    using M
    by (intro LIMSEQ_unique[OF ennreal_tendsto_cmult[OF c] LIMSEQ_INF] LIMSEQ_INF)
       (auto simp: decseq_def mult_left_mono)
qed

lemma AE_T_ev_HLD_infinite:
  fixes X :: "'s set" and r :: real
  assumes "r < 1"
  assumes r: "⋀x. x ∈ X ⟹ measure (K x) X ≤ r"
  shows "AE ω in T x. ev (HLD (- X)) ω"
proof -
  { fix x assume "x ∈ X"
    have "0 ≤ r" using r[OF ‹x ∈ X›] measure_nonneg[of "K x" X] by (blast  intro: order.trans)
    define P where "P F x = ∫⇧+y. indicator X y * (F y ⊓ 1) ∂K x" for F x
    have [measurable]: "X ∈ sets (count_space UNIV)" by auto
    have bnd: "(∫⇧+ y. indicator X y * (f y ⊓ 1) ∂K x) ≤ 1" for x f
      by (intro measure_pmf.nn_integral_le_const AE_pmfI) (auto split: split_indicator)
    have "emeasure (T x) {ω∈space (T x). alw (HLD X) ω} =
      emeasure (T x) {ω∈space (T x). gfp (λF ω. shd ω ∈ X ∧ F (stl ω)) ω}"
      by (simp add: alw_def HLD_def)
    also have "… = gfp P x"
      apply (rule emeasure_gfp)
      apply (auto intro!: order_continuous_intros inf_continuous_cmul split: split_indicator simp: P_def)
      subgoal for x f using bnd[of x f] by (auto simp: top_unique)
      subgoal for P x
        apply (subst T_eq_bind)
        apply (subst emeasure_bind[where N=S])
        apply simp
        apply (rule measurable_distr2[where M=S])
        apply (auto intro: T_subprob[THEN measurable_space] intro!: nn_integral_cong_AE AE_pmfI
            simp: emeasure_distr split: split_indicator)
        apply (simp_all add: space_stream_space T.emeasure_le_1 inf.absorb1)
        done
      apply (intro le_funI)
      apply (subst nn_integral_indicator[symmetric])
      apply simp
      apply (intro nn_integral_mono)
      apply (auto split: split_indicator)
      done
    also have "… ≤ (INF n. ennreal r ^ n)"
    proof (intro INF_greatest)
      have mono_P: "mono P"
        by (force simp: le_fun_def mono_def P_def intro!: nn_integral_mono intro: le_infI1 split: split_indicator)
      fix n show "gfp P x ≤ ennreal r ^ n"
        using ‹x ∈ X›
      proof (induction n arbitrary: x)
        case 0 then show ?case
          by (subst gfp_unfold[OF mono_P]) (auto intro!: measure_pmf.nn_integral_le_const AE_pmfI split: split_indicator simp: P_def)
      next
        case (Suc n x)
        have "gfp P x = P (gfp P) x" by (subst gfp_unfold[OF mono_P]) rule
        also have "… ≤ P (λx. ennreal r ^ n) x"
          unfolding P_def[of _ x] by (auto intro!: nn_integral_mono le_infI1 Suc split: split_indicator)
        also have "… ≤ ennreal r ^ (Suc n)"
          using Suc by (auto simp: P_def nn_integral_multc measure_pmf.emeasure_eq_measure intro!: mult_mono ennreal_leI r)
        finally show ?case .
      qed
    qed
    also have "… = 0"
      unfolding ennreal_power[OF ‹0 ≤ r›]
    proof (intro LIMSEQ_unique[OF LIMSEQ_INF])
      show "decseq (λi. ennreal (r ^ i))"
        using ‹0 ≤ r› ‹r < 1› by (auto intro!: ennreal_leI power_decreasing simp: decseq_def)
      have "(λi. ennreal (r ^ i)) ⇢ ennreal 0"
        using ‹0 ≤ r› ‹r < 1› by (intro tendsto_ennrealI LIMSEQ_power_zero) auto
      then show "(λi. ennreal (r ^ i)) ⇢ 0" by simp
    qed
    finally have *: "emeasure (T x) {ω∈space (T x). alw (HLD X) ω} = 0" by auto
    have "AE ω in T x. ev (HLD (- X)) ω"
      by (rule AE_I[OF _ *]) (auto simp: not_ev_iff not_HLD[symmetric]) }
  note * = this
  show ?thesis
    apply (clarsimp simp add: AE_T_iff[of _ x])
    subgoal for x'
      by (cases "x' ∈ X") (auto simp add: ev_Stream *)
    done
qed
  
subsection ‹Trace space with Restriction›

definition "rT x = restrict_space (T x) {ω. enabled x ω}"

lemma space_rT: "ω ∈ space (rT x) ⟷ enabled x ω"
  by (auto simp: rT_def space_restrict_space space_stream_space)

lemma Collect_enabled_S[measurable]: "Collect (enabled x) ∈ sets S"
proof -
  have "Collect (enabled x) = {ω∈space S. enabled x ω}"
    by (auto simp: space_stream_space)
  then show ?thesis
    by simp
qed

lemma space_rT_in_S: "space (rT x) ∈ sets S"
  by (simp add: rT_def space_restrict_space)

lemma sets_rT: "A ∈ sets (rT x) ⟷ A ∈ sets S ∧ A ⊆ {ω. enabled x ω}"
  by (auto simp: rT_def sets_restrict_space space_stream_space)

lemma prob_space_rT: "prob_space (rT x)"
  unfolding rT_def by (auto intro!: prob_space_restrict_space T.emeasure_eq_1_AE AE_T_enabled)

lemma measurable_force_enabled2[measurable]: "force_enabled x ∈ measurable S (rT x)"
  unfolding rT_def
  by (rule measurable_restrict_space2)
     (auto intro: measurable_force_enabled enabled_force_enabled)

lemma space_rT_not_empty[simp]: "space (rT x) ≠ {}"
  by (simp add: rT_def space_restrict_space Ex_enabled)

lemma T_eq_bind': "T x = do { y ← measure_pmf (K x) ; ω ← T y ; return S (y ## ω) }"
  apply (subst T_eq_bind)
  apply (subst bind_return_distr[symmetric])
  apply (simp_all add: space_stream_space comp_def)
  done

lemma rT_eq_bind: "rT x = do { y ← measure_pmf (K x) ; ω ← rT y ; return (rT x) (y ## ω) }"
  unfolding rT_def
  apply (subst T_eq_bind)
  apply (subst restrict_space_bind[where K=S])
  apply (rule measurable_distr2[where M=S])
  apply (auto simp del: measurable_pmf_measure1
              simp add: Ex_enabled return_restrict_space intro!: bind_cong )
  apply (subst restrict_space_bind[symmetric, where K=S])
  apply (auto simp add: Ex_enabled space_restrict_space return_cong[OF sets_T]
              intro!:  measurable_restrict_space1 measurable_compose[OF _ return_measurable]
              arg_cong2[where f=restrict_space])
  apply (subst bind_return_distr[unfolded comp_def])
  apply (simp add: space_restrict_space Ex_enabled)
  apply (simp add: measurable_restrict_space1)
  apply (rule measure_eqI)
  apply simp
  apply (subst (1 2) emeasure_distr)
  apply (auto simp: measurable_restrict_space1)
  apply (subst emeasure_restrict_space)
  apply (auto simp: space_restrict_space intro!: emeasure_eq_AE)
  using AE_T_enabled
  apply eventually_elim
  apply (simp add: space_stream_space)
  apply (rule sets_Int_pred)
  apply auto
  apply (simp add: space_stream_space)
  done

lemma snth_rT: "(λx. x !! n) ∈ measurable (rT x) (count_space (acc `` {x}))"
proof -
  have "⋀ω. enabled x ω ⟹ (x, ω !! n) ∈ acc"
  proof (induction n arbitrary: x)
    case (Suc n) from Suc.prems Suc.IH[of "shd ω" "stl ω"] show ?case
      by (auto simp: enabled.simps[of x ω] intro: rtrancl_trans)
  qed (auto elim: enabled.cases)
  moreover
  { fix X :: "'s set"
    have [measurable]: "X ∈ count_space UNIV" by simp
    have *: "(λx. x !! n) -` X ∩ space (rT x) =  {ω∈space S. ω !! n ∈ X ∧ enabled x ω}"
      by (auto simp: space_stream_space space_rT)
    have "(λx. x !! n) -` X ∩ space (rT x) ∈ sets S"
      unfolding * by measurable }
  ultimately show ?thesis
    by (auto simp: measurable_def space_rT sets_rT)
qed

subsection ‹Bisimulation›

lemma T_coinduct[consumes 1, case_names prob sets cont]:
  assumes "R x M"
  assumes prob: "⋀x M. R x M ⟹ prob_space M"
    and sets: "⋀x M. R x M ⟹ sets M = sets S"
    and cont': "⋀x M. R x M ⟹ ∃M'. (∀y∈K x. R y (M' y)) ∧ (∀y. sets (M' y) = S ∧ prob_space (M' y)) ∧
      M = (measure_pmf (K x) ⤜ (λy. distr (M' y) S ((##) y)))"
  shows "T x = M"
  using ‹R x M›
proof (coinduction arbitrary: x M rule: measure_eq_stream_space_coinduct)
  case left then show ?case using T.prob_space_axioms[of x] sets_T[of x] by (auto simp: space_prob_algebra)
next
  case (right M) with prob[of M] sets[of M] show ?case by (auto simp: space_prob_algebra)
next
  case (cont x M) with cont'[OF cont] obtain M' where *:
    "(∀y∈K x. R y (M' y))"
    "(∀y. sets (M' y) = S ∧ prob_space (M' y))"
    "M = (measure_pmf (K x) ⤜ (λy. distr (M' y) S ((##) y)))"
    by auto
  show ?case
    apply (rule exI[of _ T])
    apply (rule exI[of _ M'])
    apply (rule exI[of _ "K x"])
    using * T.prob_space_axioms sets_T[of x]
    apply (auto simp: space_prob_algebra intro: T_eq_bind)
    done
qed

lemma T_bisim:
  assumes M: "⋀x. prob_space (M x)" "⋀x. sets (M x) = sets S"
    and M_eq: "⋀x. M x = (measure_pmf (K x) ⤜ (λs. distr (M s) S ((##) s)))"
  shows "T = M"
proof
  fix x show "T x = M x"
  proof (coinduction arbitrary: x rule: T_coinduct)
    case (cont x) then show ?case
      apply (intro exI[of _ M])
      apply (subst M_eq[of x])
      apply (simp add: M)
      done
  qed fact+
qed

lemma T_subprob'[measurable]: "T ∈ measurable (count_space UNIV) (subprob_algebra S)"
  by (auto intro!: space_bind simp: space_subprob_algebra) unfold_locales

lemma T_subprob''[simp]: "T a ∈ space (subprob_algebra S)"
  using measurable_space[OF T_subprob', of a] by simp

lemma AE_not_suntil_coinduct [consumes 1, case_names ψ φ]:
  assumes "P s"
  assumes ψ: "⋀s. P s ⟹ s ∉ ψ"
  assumes φ: "⋀s t. P s ⟹ s ∈ φ ⟹ t ∈ K s ⟹ P t"
  shows "AE ω in T s. not (HLD φ suntil HLD ψ) (s ## ω)"
proof -
  { fix ω have "¬ (HLD φ suntil HLD ψ) (s ## ω) ⟷
      (∀n. ¬ ((λR. HLD ψ or (HLD φ aand nxt R)) ^^ n) ⊥ (s ## ω))"
      unfolding suntil_def
      by (subst sup_continuous_lfp)
         (auto simp add: sup_continuous_def) }
  moreover
  { fix n from ‹P s› have "AE ω in T s. ¬ ((λR. HLD ψ or (HLD φ aand nxt R)) ^^ n) ⊥ (s ## ω)"
    proof (induction n arbitrary: s)
      case (Suc n) then show ?case
        apply (subst AE_T_iff)
        apply (rule measurable_compose[OF measurable_Stream, where M1="count_space UNIV"])
        apply measurable
        apply simp
        apply (auto simp: bot_fun_def intro!: AE_impI dest: φ ψ)
        done
    qed simp }
  ultimately show ?thesis
    by (simp add: AE_all_countable)
qed

lemma AE_not_suntil_coinduct_strong [consumes 1, case_names ψ φ]:
  assumes "P s"
  assumes P_ψ: "⋀s. P s ⟹ s ∉ ψ"
  assumes P_φ: "⋀s t. P s ⟹ s ∈ φ ⟹ t ∈ K s ⟹ P t ∨
    (AE ω in T t. not (HLD φ suntil HLD ψ) (t ## ω))"
  shows "AE ω in T s. not (HLD φ suntil HLD ψ) (s ## ω)" (is "?nuntil s")
proof -
  have "P s ∨ ?nuntil s"
    using ‹P s› by auto
  then show ?thesis
  proof (coinduction arbitrary: s rule: AE_not_suntil_coinduct)
    case (φ t s) then show ?case
      by (auto simp: AE_T_iff[of _ s] suntil_Stream[of _ _ s] dest: P_φ)
  qed (auto simp: suntil_Stream dest: P_ψ)
qed

end

subsection ‹Reward Structure on Markov Chains›

locale MC_with_rewards = MC_syntax K for K :: "'s ⇒ 's pmf" +
  fixes ι :: "'s ⇒ 's ⇒ ennreal" and ρ :: "'s ⇒ ennreal"
  assumes ι_nonneg: "⋀s t. 0 ≤ ι s t" and ρ_nonneg: "⋀s. 0 ≤ ρ s"
  assumes measurable_ι[measurable]: "(λ(a, b). ι a b) ∈ borel_measurable (count_space UNIV ⨂⇩M count_space UNIV)"
begin

definition reward_until :: "'s set ⇒ 's ⇒ 's stream ⇒ ennreal" where
  "reward_until X = lfp (λF s ω. if s ∈ X then 0 else ρ s + ι s (shd ω) + (F (shd ω) (stl ω)))"

lemma measurable_ρ[measurable]: "ρ ∈ borel_measurable (count_space UNIV)"
  by simp

lemma measurable_reward_until[measurable (raw)]:
  assumes [measurable]: "f ∈ measurable M (count_space UNIV)"
  assumes [measurable]: "g ∈ measurable M S"
  shows "(λx. reward_until X (f x) (g x)) ∈ borel_measurable M"
proof -
  let ?F = "λF (s, ω). if s ∈ X then 0 else ρ s + ι s (shd ω) + (F (shd ω, stl ω))"
  { fix s ω
    have "reward_until X s ω = lfp ?F (s, ω)"
      unfolding reward_until_def lfp_pair[symmetric] .. }
  note * = this

  have [measurable]: "lfp ?F ∈ borel_measurable (count_space UNIV ⨂⇩M S)"
  proof (rule borel_measurable_lfp)
    fix f :: "('s × 's stream) ⇒ ennreal"
    assume [measurable]: "f ∈ borel_measurable (count_space UNIV ⨂⇩M S)"
    show "?F f ∈ borel_measurable (count_space UNIV ⨂⇩M S)"
      unfolding split_beta'
      apply (intro measurable_If)
      apply measurable []
      apply measurable []
      apply (rule predE)
      apply (rule measurable_compose[OF measurable_fst])
      apply measurable []
      done
  qed (auto intro!: ι_nonneg ρ_nonneg order_continuous_intros)
  show ?thesis
    unfolding * by measurable
qed

lemma continuous_reward_until:
  "sup_continuous (λF s ω. if s ∈ X then 0 else ρ s + ι s (shd ω) + (F (shd ω) (stl ω)))"
  by (intro ι_nonneg ρ_nonneg order_continuous_intros) (auto simp: sup_continuous_def image_comp)

lemma
  shows reward_until_unfold: "reward_until X s ω =
        (if s ∈ X then 0 else ρ s + ι s (shd ω) + reward_until X (shd ω) (stl ω))"
      (is ?unfold)
proof -
  let ?F = "λF s ω. if s ∈ X then 0 else ρ s + ι s (shd ω) + (F (shd ω) (stl ω))"
  { fix s ω have "reward_until X s ω = ?F (reward_until X) s ω"
      unfolding reward_until_def
      apply (subst lfp_unfold)
      apply (rule continuous_reward_until[THEN sup_continuous_mono, of X])
      apply rule
      done }
  note step = this
  show ?unfold
    by (subst step) (auto intro!: arg_cong2[where f="(+)"])
qed

lemma reward_until_simps[simp]:
  shows "s ∈ X ⟹ reward_until X s ω = 0"
    and "s ∉ X ⟹ reward_until X s ω = ρ s + ι s (shd ω) + reward_until X (shd ω) (stl ω)"
  unfolding reward_until_unfold[of X s ω] by simp_all

lemma reward_until_SCons[simp]:
  "reward_until X s (t ## ω) = (if s ∈ X then 0 else ρ s + ι s t + reward_until X t ω)"
  by simp

lemma nn_integral_reward_until_finite:
  assumes [simp]: "finite (acc `` {s})" (is "finite (?R `` {s})")
  assumes ρ: "⋀t. (s, t) ∈ acc_on (-H) ⟹ ρ t < ∞"
  assumes ι: "⋀t t'. (s, t) ∈ acc_on (-H) ⟹ t' ∈ K t ⟹ ι t t' < ∞"
  assumes ev: "AE ω in T s. ev (HLD H) ω"
  shows "(∫⇧+ ω. reward_until H s ω ∂T s) ≠ ∞"
proof cases
  assume "s ∈ H" then show ?thesis
    by simp
next
  assume "s ∉  H"
  let ?L = "acc_on (-H)"
  define M where "M = Max ((λ(s, t). ρ s + ι s t) ` (SIGMA t:?L``{s}. K t))"
  have "?L ⊆ ?R"
    by (intro rtrancl_mono) auto
  with ‹s ∉ H› have subset: "(SIGMA t:?L``{s}. K t) ⊆ (?R``{s} × ?R``{s})"
    by (auto intro: rtrancl_into_rtrancl elim: rtrancl.cases)
  then have [simp, intro!]: "finite ((λ(s, t). ρ s + ι s t) ` (SIGMA t:?L``{s}. K t))"
    by (intro finite_imageI) (auto dest: finite_subset)
  { fix t t' assume "(s, t) ∈ ?L" "t ∉ H" "t' ∈ K t"
    then have "(t, t') ∈ (SIGMA t:?L``{s}. K t)"
      by (auto intro: rtrancl_into_rtrancl)
    then have "ρ t + ι t t' ≤ M"
      unfolding M_def by (intro Max_ge) auto }
  note le_M = this

  have fin_L: "finite (?L `` {s})"
    by (intro finite_subset[OF _ assms(1)] Image_mono ‹?L ⊆ ?R› order_refl)

  have "M < ∞"
    unfolding M_def
  proof (subst Max_less_iff, safe)
    show "(SIGMA x:?L `` {s}. set_pmf (K x)) = {} ⟹ False"
      using ‹s ∉ H› by (auto simp add: Sigma_empty_iff set_pmf_not_empty)
    fix t t' assume "(s, t) ∈ ?L" "t' ∈ K t" then show "ρ t + ι t t' < ∞"
      using ρ[of t] ι[of t t'] by simp
  qed

  from set_pmf_not_empty[of "K s"] obtain t where "t ∈ K s"
    by auto
  with le_M[of s t] have "0 ≤ M"
    using set_pmf_not_empty[of "K s"] ‹s ∉ H› le_M[of s] ι_nonneg[of s] ρ_nonneg[of s]
    by (intro order_trans[OF _ le_M]) auto

  have "AE ω in T s. reward_until H s ω ≤ M * sfirst (HLD H) (s ## ω)"
    using ev AE_T_enabled
  proof eventually_elim
    fix ω assume "ev (HLD H) ω" "enabled s ω"
    moreover define t where "t = s"
    ultimately have "ev (HLD H) ω" "enabled t ω" "t ∈ ?L``{s}"
      by auto
    then show "reward_until H t ω ≤ M * sfirst (HLD H) (t ## ω)"
    proof (induction arbitrary: t rule: ev_induct_strong)
      case (base ω t) then show ?case
        by (auto simp: HLD_iff sfirst_Stream elim: enabled.cases intro: le_M)
    next
      case (step ω t) from step.IH[of "shd ω"] step.prems step.hyps show ?case
        by (auto simp add: HLD_iff enabled.simps[of t] distrib_left sfirst_Stream
                           reward_until_simps[of t]
                 simp del: reward_until_simps
                 intro!: add_mono le_M intro: rtrancl_into_rtrancl)
    qed
  qed
  then have "(∫⇧+ω. reward_until H s ω ∂T s) ≤ (∫⇧+ω. M * sfirst (HLD H) (s ## ω) ∂T s)"
    by (rule nn_integral_mono_AE)
  also have "… < ∞"
    using ‹0 ≤ M› ‹M < ∞› nn_integral_sfirst_finite[OF fin_L ev]
    by (simp add: nn_integral_cmult  less_top[symmetric] ennreal_mult_eq_top_iff)
  finally show ?thesis
    by simp
qed

end

subsection ‹Bisimulation on a relation›

definition rel_set_strong :: "('a ⇒ 'b ⇒ bool) ⇒ 'a set ⇒ 'b set ⇒ bool"
  where "rel_set_strong R A B ⟷ (∀x y. R x y ⟶ (x ∈ A ⟷ y ∈ B))"

lemma T_eq_rel_half[consumes 4, case_names prob sets cont]:
  fixes R :: "'s ⇒ 't ⇒ bool" and f :: "'s ⇒ 't" and S :: "'s set"
  assumes R_def: "⋀s t. R s t ⟷ (s ∈ S ∧ f s = t)"
  assumes A[measurable]: "A ∈ sets (stream_space (count_space UNIV))"
    and B[measurable]: "B ∈ sets (stream_space (count_space UNIV))"
    and AB: "rel_set_strong (stream_all2 R) A B" and KL: "rel_fun R (rel_pmf R) K L" and xy: "R x y"
  shows "MC_syntax.T K x A = MC_syntax.T L y B"
proof -
  interpret K: MC_syntax K by unfold_locales
  interpret L: MC_syntax L by unfold_locales

  have "x ∈ S" using ‹R x y› by (auto simp: R_def)

  define g where "g t = (SOME s. R s t)" for t
  have measurable_g: "g ∈ count_space UNIV →⇩M count_space UNIV" by auto
  have g: "R i j ⟹ R (g j) j" for i j
    unfolding g_def by (rule someI)
  
  have K_subset: "x ∈ S ⟹ K x ⊆ S" for x
    using KL[THEN rel_funD, of x "f x", THEN rel_pmf_imp_rel_set] by (auto simp: rel_set_def R_def)

  have in_S: "AE ω in K.T x. ω ∈ streams S"
    using K.AE_T_enabled
  proof eventually_elim 
    case (elim ω) with ‹x ∈ S› show ?case
      apply (coinduction arbitrary: x ω)
      subgoal for x ω using K_subset by (cases ω) (auto simp: K.enabled_Stream)
      done
  qed

  have L_eq: "L y = map_pmf f (K x)" if xy: "R x y" for x y
  proof -
    have "rel_pmf (λx y. x = y) (map_pmf f (K x)) (L y)"
      using KL[THEN rel_funD, OF xy] by (auto intro: pmf.rel_mono_strong simp: R_def pmf.rel_map)
    then show ?thesis unfolding pmf.rel_eq by simp
  qed

  let ?D = "λx. distr (K.T x) K.S (smap f)"
  have prob_space_D: "?D x ∈ space (prob_algebra K.S)" for x 
    by (auto simp: space_prob_algebra K.T.prob_space_distr)

  have D_eq_D: "?D x = ?D x'" if "R x y" "R x' y" for x x' y
  proof (rule stream_space_eq_sstart)
    define A where "A = K.acc `` {x, x'}"
    have x_A: "x ∈ A" "x' ∈ A" by (auto simp: A_def)
    let ?Ω = "f ` A"
    show "countable ?Ω"
      unfolding A_def by (intro countable_image K.countable_acc) auto
    show "prob_space (?D x)" "prob_space (?D x')" by (auto intro!: K.T.prob_space_distr)
    show "sets (?D x) = sets L.S" "sets (?D x') = sets L.S" by auto
    have AE_streams: "AE x in ?D x''. x ∈ streams ?Ω" if "x'' ∈ A" for x''
      apply (simp add: space_stream_space streams_sets AE_distr_iff)
      using K.AE_T_reachable[of x''] unfolding alw_HLD_iff_streams
    proof eventually_elim
      fix s assume "s ∈ streams (K.acc `` {x''})"
      moreover have "K.acc `` {x''} ⊆ A"
        using ‹x'' ∈ A› by (auto simp: A_def Image_def intro: rtrancl_trans)
      ultimately show "smap f s ∈ streams (f ` A)"
        by (auto intro: smap_streams)
    qed
    with x_A show "AE x in ?D x'. x ∈ streams ?Ω" "AE x in ?D x. x ∈ streams ?Ω"
      by auto
    from ‹x ∈ A› ‹x' ∈ A› that show "?D x (sstart (f ` A) xs) = ?D x' (sstart (f ` A) xs)" for xs
    proof (induction xs arbitrary: x x' y)
      case Nil
      moreover have "?D x (streams (f ` A)) = 1" if "x ∈ A" for x
        using AE_streams[of x] that
        by (intro prob_space.emeasure_eq_1_AE[OF K.T.prob_space_distr]) (auto simp: streams_sets)
      ultimately show ?case by simp
    next
      case (Cons z zs x x' y)
      have "rel_pmf (R OO R¯¯) (K x) (K x')"
        using KL[THEN rel_funD, OF Cons(4)] KL[THEN rel_funD, OF Cons(5)]
        unfolding pmf.rel_compp pmf.rel_flip by auto
      then obtain p :: "('s × 's) pmf" where p: "⋀a b. (a, b) ∈ p ⟹ (R OO R¯¯) a b" and
        eq: "map_pmf fst p = K x" "map_pmf snd p = K x'"
        by (auto simp: pmf.in_rel)
      let ?S = "stream_space (count_space UNIV)"
      have *: "(##) y -` smap f -` sstart (f ` A) (z # zs) = (if f y = z then smap f -` sstart (f ` A) zs else {})" for y z zs
        by auto
      have **: "?D x (sstart (f ` A) (z # zs)) = (∫⇧+ y'. (if f y' = z then ?D y' (sstart (f ` A) zs) else 0) ∂K x)" for x
        apply (simp add: emeasure_distr)
        apply (subst K.T_eq_bind)
        apply (subst emeasure_bind[where N="?S"])
           apply simp
          apply (rule measurable_distr2[where M="?S"])
           apply measurable
        apply (intro nn_integral_cong_AE AE_pmfI)
        apply (auto simp add: emeasure_distr)
        apply (simp_all add: * space_stream_space)
        done
      have fst_A: "fst ab ∈ A" if "ab ∈ p" for ab
      proof -
        have "fst ab ∈ K x" using ‹ab ∈ p› set_map_pmf [of fst p] by (auto simp: eq)
        with ‹x ∈ A› show "fst ab ∈ A"
          by (auto simp: A_def intro: rtrancl.rtrancl_into_rtrancl)
      qed
      have snd_A: "snd ab ∈ A" if "ab ∈ p" for ab
      proof -
        have "snd ab ∈ K x'" using ‹ab ∈ p› set_map_pmf [of snd p] by (auto simp: eq)
        with ‹x' ∈ A› show "snd ab ∈ A"
          by (auto simp: A_def intro: rtrancl.rtrancl_into_rtrancl)
      qed
      show ?case
        unfolding ** eq[symmetric] nn_integral_map_pmf
        apply (intro nn_integral_cong_AE AE_pmfI)
        subgoal for ab using p[of "fst ab" "snd ab"] by (auto simp: R_def intro!: Cons(1) fst_A snd_A)
        done
    qed
  qed

  have L_eq_D: "L.T y = ?D x"
    using ‹R x y›
  proof (coinduction arbitrary: x y rule: L.T_coinduct)
    case (cont x y)
    then have Kx_Ly: "rel_pmf R (K x) (L y)"
      by (rule KL[THEN rel_funD])
    then have *: "y' ∈ L y ⟹ ∃x'∈K x. R x' y'" for y'
      by (auto dest!: rel_pmf_imp_rel_set simp: rel_set_def)
    have **: "y' ∈ L y ⟹ R (g y') y'" for y'
      using *[of y'] unfolding g_def by (auto intro: someI)

    have D_SCons_eq_D_D: "distr (K.T i) K.S (λx. z ## smap f x) = distr (?D i) K.S (λx. z ## x)" for i z
      by (subst distr_distr) (auto simp: comp_def)
    have D_eq_D_gi: "?D i = ?D (g (f i))" if i: "i ∈ K x" for i
    proof -
      obtain j where "j ∈ L y" "R i j" "f i = j"
        using Kx_Ly i by (force dest!: rel_pmf_imp_rel_set simp: rel_set_def R_def)
      then show ?thesis
        by (auto intro!: D_eq_D[OF ‹R i j›] g)
    qed

    have ***: "?D x = measure_pmf (L y) ⤜ (λy. distr (?D (g y)) K.S ((##) y))"
      apply (subst K.T_eq_bind)
      apply (subst distr_bind[of _ _ K.S])
         apply (rule measurable_distr2[of _  _ "K.S"])
          apply (simp_all add: Pi_iff)
      apply (simp add: distr_distr comp_def L_eq[OF cont] map_pmf_rep_eq)
      apply (subst bind_distr[where K=K.S])
         apply measurable []
        apply (rule measurable_distr2[of _  _ "K.S"])
        apply measurable []
        apply (rule measurable_compose[OF measurable_g])
        apply measurable []
       apply simp
      apply (rule bind_measure_pmf_cong[where N="K.S"])
        apply (auto simp: space_subprob_algebra space_stream_space intro!: K.T.subprob_space_distr)
        unfolding D_SCons_eq_D_D D_eq_D_gi ..
    show ?case
      by (intro exI[of _ "λt. distr (K.T (g t)) (stream_space (count_space UNIV)) (smap f)"])
         (auto simp add: K.T.prob_space_distr *** dest: **)
  qed (auto intro: K.T.prob_space_distr)

  have "stream_all2 R s t ⟷ (s ∈ streams S ∧ smap f s = t)" for s t 
  proof safe
    show "stream_all2 R s t ⟹ s ∈ streams S"
      apply (coinduction arbitrary: s t)
      subgoal for s t by (cases s; cases t) (auto simp: R_def)
      done
    show "stream_all2 R s t ⟹ smap f s = t"
      apply (coinduction arbitrary: s t)
      subgoal for s t by (cases s; cases t) (auto simp: R_def)
      done
  qed (auto intro!: stream.rel_refl_strong simp: stream.rel_map R_def streams_iff_sset)
  then have "ω ∈ streams S ⟹ ω ∈ A ⟷ smap f ω ∈ B" for ω
    using AB by (auto simp: rel_set_strong_def)
  with in_S have "K.T x A = K.T x (smap f -` B ∩ space (K.T x))"
    by (auto intro!: emeasure_eq_AE streams_sets)
  also have "… = (distr (K.T x) K.S (smap f)) B"
    by (intro emeasure_distr[symmetric]) auto
  also have "… = (L.T y) B" unfolding L_eq_D ..
  finally show ?thesis .
qed

subsection ‹Product Construction›

locale MC_pair =
  K1: MC_syntax K1 + K2: MC_syntax K2 for K1 K2
begin

definition "Kp ≡ λ(a, b). pair_pmf (K1 a) (K2 b)"

sublocale MC_syntax Kp .

definition
  "szip⇩E a b ≡ λ(ω1, ω2). szip (K1.force_enabled a ω1) (K2.force_enabled b ω2)"

lemma szip_rT[measurable]: "(λ(ω1, ω2). szip ω1 ω2) ∈ measurable (K1.rT x1 ⨂⇩M K2.rT x2) S"
proof (rule measurable_stream_space2)
  fix n
  have "(λx. (case x of (ω1, ω2) ⇒ szip ω1 ω2) !! n) = (λω. (fst ω !! n, snd ω !! n))"
    by auto
  also have "… ∈ measurable (K1.rT x1 ⨂⇩M K2.rT x2) (count_space UNIV)"
    apply (rule measurable_compose_countable'[OF _ measurable_compose[OF measurable_fst K1.snth_rT, of n]])
    apply (rule measurable_compose_countable'[OF _ measurable_compose[OF measurable_snd K2.snth_rT, of n]])
    apply (auto intro!: K1.countable_acc K2.countable_acc)
    done
  finally show "(λx. (case x of (ω1, ω2) ⇒ szip ω1 ω2) !! n) ∈ measurable (K1.rT x1 ⨂⇩M K2.rT x2) (count_space UNIV)"
    .
qed

lemma measurable_szipE[measurable]: "szip⇩E a b ∈ measurable (K1.S ⨂⇩M K2.S) S"
  unfolding szip⇩E_def by measurable

lemma T_eq_prod: "T = (λ(x1, x2). do { ω1 ← K1.T x1 ; ω2 ← K2.T x2 ; return S (szip⇩E x1 x2 (ω1, ω2)) })"
  (is "_ = ?B")
proof (rule T_bisim)
  have T1x: "⋀x. subprob_space (K1.T x)"
    by (rule prob_space_imp_subprob_space) unfold_locales

  interpret T12: pair_prob_space "K1.T x" "K2.T y" for x y
    by unfold_locales
  interpret T1K2: pair_prob_space "K1.T x" "K2 y" for x y
    by unfold_locales

  let ?P = "λx1 x2. K1.T x1 ⨂⇩M K2.T x2"

  fix x show "prob_space (?B x)"
    by (auto simp: space_stream_space split: prod.splits
                intro!: prob_space.prob_space_bind prob_space_return
                        measurable_bind[where N=S] measurable_compose[OF _ return_measurable] AE_I2)
       unfold_locales

  show "sets (?B x) = sets S"
    by (simp split: prod.splits add: measurable_bind[where N=S] sets_bind[where N=S] space_stream_space)

  obtain a b where x_eq: "x = (a, b)"
    by (cases x) auto
  show "?B x = (measure_pmf (Kp x) ⤜ (λs. distr (?B s) S ((##) s)))"
    unfolding x_eq
    apply (subst K1.T_eq_bind')
    apply (subst K2.T_eq_bind')
    apply (auto
         simp add: space_stream_space bind_assoc[where R=S and N=S] bind_return_distr[symmetric]
                   Kp_def T1K2.bind_rotate[where N=S] split_beta' set_pair_pmf space_subprob_algebra
                   bind_pair_pmf[of "case_prod M" for M, unfolded split, symmetric, where N=S] szip⇩E_def
                   stream_eq_Stream_iff bind_return[where N=S] space_bind[where N=S]
         simp del: measurable_pmf_measure1
         intro!: bind_measure_pmf_cong[where N=S] subprob_space_bind[where N=S] subprob_space_measure_pmf
                 T1x bind_cong[where M="MC_syntax.T K x" for K x] arg_cong2[where f=return])
    done
qed

lemma nn_integral_pT:
  fixes f assumes [measurable]: "f ∈ borel_measurable S"
  shows "(∫⇧+ω. f ω ∂T (x, y)) = (∫⇧+ω1. ∫⇧+ω2. f (szip⇩E x y (ω1, ω2)) ∂K2.T y ∂K1.T x)"
  by (simp add: nn_integral_bind[where B=S] nn_integral_return in_S T_eq_prod)

lemma prod_eq_prob_T:
  assumes [measurable]: "Measurable.pred K1.S P1" "Measurable.pred K2.S P2"
  shows "𝒫(ω in K1.T x1. P1 ω) * 𝒫(ω in K2.T x2. P2 ω) =
    𝒫(ω in T (x1, x2). P1 (smap fst ω) ∧ P2 (smap snd ω))"
proof -
  have "𝒫(ω in T (x1, x2). P1 (smap fst ω) ∧ P2 (smap snd ω)) =
    (∫ x. ∫ xa. indicator {ω ∈ space S. P1 (smap fst ω) ∧ P2 (smap snd ω)} (szip⇩E x1 x2 (x, xa)) ∂MC_syntax.T K2 x2 ∂MC_syntax.T K1 x1)"
    by (subst T_eq_prod)
       (simp add: K1.T.measure_bind[where N=S] K2.T.measure_bind[where N=S] measure_return)
  also have "... = (∫ω1. ∫ω2. indicator {ω∈space K1.S. P1 ω} ω1 * indicator {ω∈space K2.S. P2 ω} ω2 ∂K2.T x2 ∂K1.T x1)"
    apply (intro integral_cong_AE)
    apply measurable
    using K1.AE_T_enabled
    apply eventually_elim
    apply (intro integral_cong_AE)
    apply measurable
    using K2.AE_T_enabled
    apply eventually_elim
    apply (auto simp: space_stream_space szip⇩E_def K1.force_enabled K2.force_enabled
                      smap_szip_snd[where g="λx. x"] smap_szip_fst[where f="λx. x"]
                split: split_indicator)
    done
  also have "… = 𝒫(ω in K1.T x1. P1 ω) * 𝒫(ω in K2.T x2. P2 ω)"
    by simp
  finally show ?thesis ..
qed

end

end